18
Basic Laws Basic Definitions mportant Derivations Examples + Maxwell’s Equations Fields & Potentials Energies, Currents & Power Fields, Motion & Polarizations
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Basic Laws
Basic Definitions
Important Derivations
Examples
+Maxwellrsquos Equations
Fields ampPotentials
Energies Currentsamp Power
Fields Motionamp Polarizations
Gauss Lawbull Gauss Law (a FUNDAMENTAL Law)
The net electric flux through any closed surface is proportional to the charge enclosed by that surface
enclosedqSdE 00
bull How to Applyndash Useful in finding E when the physical situation has SYMMETRYndash CHOOSE a closed surface such that the integral is TRIVIAL
raquo Direction surface must be chosen such that E is known to be either parallel or perpendicular to each piece of the surface
raquo Magnitude surface must be chosen such that E has the same value at all points on the surface when E is perpendicular to the surface
raquo Therefore bring E outside of the integralbull Most of our results for E fields around points lines
planes were obtained this way
Gaussrsquo Law qSdE
0
ErSdE 200 4
LHSRHS q = ALL charge inside radius r
204
1rqE
rLESdE 200
LHSRHS q = ALL charge inside radius
r length Lr
E02
AESdE 200
LHS
RHS q = ALL charge inside cylinder=A
02
E
bull Spherical Symmetry
bull Cylindrical Symmetry
bull Planar Symmetry
Use superposition of these results
Gaussrsquo Law Example
Consider
bull What is the surface charge density L on the left side of the conducting slab
1 2
EA
A
L RTwo non-conducting infinite sheets with surface charge densities 1=+3 Cm2 and 2 = +5 Cm2 and an uncharged conducting slab
E=0
1 Use SUPERPOSITION of Electric Fields
On Far Left = 120 ( -1 - 2 - L - R )
At X = 120 ( +1 - 2 - L - R )
In Red = 120 (1 - 2 + L - R ) = 0
On Far Right = 12e0 (1 + 2 + L + R )
2 Uncharged L + R = 0 Add L - R = 1 - 2 Get L = (1 - 2)2
E
E
E
E
E
E
Electric Potential Differencebull Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field
bull Since there will be a force on the charge due to a certain amount of work W will have to be done to accomplish this task We define the electric potential difference as
bull Since the force we have to exert must just cancel the electric force
A B
q0
0qWVV AB
AB
B
A
ABAB ldE
qWVV
0EqF
E E
E
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Gauss Lawbull Gauss Law (a FUNDAMENTAL Law)
The net electric flux through any closed surface is proportional to the charge enclosed by that surface
enclosedqSdE 00
bull How to Applyndash Useful in finding E when the physical situation has SYMMETRYndash CHOOSE a closed surface such that the integral is TRIVIAL
raquo Direction surface must be chosen such that E is known to be either parallel or perpendicular to each piece of the surface
raquo Magnitude surface must be chosen such that E has the same value at all points on the surface when E is perpendicular to the surface
raquo Therefore bring E outside of the integralbull Most of our results for E fields around points lines
planes were obtained this way
Gaussrsquo Law qSdE
0
ErSdE 200 4
LHSRHS q = ALL charge inside radius r
204
1rqE
rLESdE 200
LHSRHS q = ALL charge inside radius
r length Lr
E02
AESdE 200
LHS
RHS q = ALL charge inside cylinder=A
02
E
bull Spherical Symmetry
bull Cylindrical Symmetry
bull Planar Symmetry
Use superposition of these results
Gaussrsquo Law Example
Consider
bull What is the surface charge density L on the left side of the conducting slab
1 2
EA
A
L RTwo non-conducting infinite sheets with surface charge densities 1=+3 Cm2 and 2 = +5 Cm2 and an uncharged conducting slab
E=0
1 Use SUPERPOSITION of Electric Fields
On Far Left = 120 ( -1 - 2 - L - R )
At X = 120 ( +1 - 2 - L - R )
In Red = 120 (1 - 2 + L - R ) = 0
On Far Right = 12e0 (1 + 2 + L + R )
2 Uncharged L + R = 0 Add L - R = 1 - 2 Get L = (1 - 2)2
E
E
E
E
E
E
Electric Potential Differencebull Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field
bull Since there will be a force on the charge due to a certain amount of work W will have to be done to accomplish this task We define the electric potential difference as
bull Since the force we have to exert must just cancel the electric force
A B
q0
0qWVV AB
AB
B
A
ABAB ldE
qWVV
0EqF
E E
E
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Gaussrsquo Law qSdE
0
ErSdE 200 4
LHSRHS q = ALL charge inside radius r
204
1rqE
rLESdE 200
LHSRHS q = ALL charge inside radius
r length Lr
E02
AESdE 200
LHS
RHS q = ALL charge inside cylinder=A
02
E
bull Spherical Symmetry
bull Cylindrical Symmetry
bull Planar Symmetry
Use superposition of these results
Gaussrsquo Law Example
Consider
bull What is the surface charge density L on the left side of the conducting slab
1 2
EA
A
L RTwo non-conducting infinite sheets with surface charge densities 1=+3 Cm2 and 2 = +5 Cm2 and an uncharged conducting slab
E=0
1 Use SUPERPOSITION of Electric Fields
On Far Left = 120 ( -1 - 2 - L - R )
At X = 120 ( +1 - 2 - L - R )
In Red = 120 (1 - 2 + L - R ) = 0
On Far Right = 12e0 (1 + 2 + L + R )
2 Uncharged L + R = 0 Add L - R = 1 - 2 Get L = (1 - 2)2
E
E
E
E
E
E
Electric Potential Differencebull Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field
bull Since there will be a force on the charge due to a certain amount of work W will have to be done to accomplish this task We define the electric potential difference as
bull Since the force we have to exert must just cancel the electric force
A B
q0
0qWVV AB
AB
B
A
ABAB ldE
qWVV
0EqF
E E
E
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Gaussrsquo Law Example
Consider
bull What is the surface charge density L on the left side of the conducting slab
1 2
EA
A
L RTwo non-conducting infinite sheets with surface charge densities 1=+3 Cm2 and 2 = +5 Cm2 and an uncharged conducting slab
E=0
1 Use SUPERPOSITION of Electric Fields
On Far Left = 120 ( -1 - 2 - L - R )
At X = 120 ( +1 - 2 - L - R )
In Red = 120 (1 - 2 + L - R ) = 0
On Far Right = 12e0 (1 + 2 + L + R )
2 Uncharged L + R = 0 Add L - R = 1 - 2 Get L = (1 - 2)2
E
E
E
E
E
E
Electric Potential Differencebull Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field
bull Since there will be a force on the charge due to a certain amount of work W will have to be done to accomplish this task We define the electric potential difference as
bull Since the force we have to exert must just cancel the electric force
A B
q0
0qWVV AB
AB
B
A
ABAB ldE
qWVV
0EqF
E E
E
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Electric Potential Differencebull Suppose charge q0 is moved from pt
A to pt B through a region of space described by electric field
bull Since there will be a force on the charge due to a certain amount of work W will have to be done to accomplish this task We define the electric potential difference as
bull Since the force we have to exert must just cancel the electric force
A B
q0
0qWVV AB
AB
B
A
ABAB ldE
qWVV
0EqF
E E
E
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Potential from N chargesThe potential from a collection of N charges is just the algebraic sum of the potential due to each charge separately
xr1
r2 r3
q1
q3
q2
rr
r
N
nn
rr
r
ldEldErV1
)(
N
n n
nN
nn r
qrVrV101 4
1)()(
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
allows us to calculate the potential function V everywhere (keep in mind we often define VA = 0 at some convenient place)
If we know the electric field E everywhere
1
allows us to calculate the electric field E everywhere
If we know the potential function V everywhere
bull Units for Potential 1 JouleCoul = 1 VOLT
The Bottom Line
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Q R1
R2
R3
P
A charge of Q = 4nC is placed on a solid conducting sphere of radius R1 = 5cm and surrounded by a concentric conducting solid shell with inner radius R2 = 20cm and outer radius R3 = 25cm and no net charge
Now we connect the inner sphere to the outer shell with a conducting wireWhat is the potential at R1 (assuming V = 0 at infinity)a kQ R1
b kQ R2
c kQ R3
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
C1 = 1 F
C2 = 2 F
C3 = 3 F12 v
V = 0
b
The potential along the bottom wire is defined to be zero
The voltage Vb at the point labeled b isa 10 V b 546 Vc 40 Vd 20 Ve 017 V
Assume that all capacitors are uncharged before the circuit is assembled
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
The north end of a magnet is very slowly brought close to a block of some unknown substance If the block is attracted by the magnet the substance could equally well be either a hard ferromagnet or diamagnetic (Assume the attraction exists even after all possible induced surface currents have died away if it helps remember that water is diamagnetic and liquid oxygen is paramagnetic)
NS F
a Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
An infinitely long coaxial cable consists of a solid conducting cylinder of radius R1 = 1mm and a hollow conducting shell with inner radius R2=2mm and outer radius R3=3mm The cable is aligned along the z axis and centered at x=y=0 The inner conductor carries a uniformly distributed current I1 = 2 A in the +z direction (out of the page) and the conducting shell carries a uniformly distributed current of I2 = 5 A in the ndashz direction (into the page)
R3
Px
y
R1 = 1 mmR2 = 2 mmR3 = 3 mm
R1
R2
The magnetic field at the point P is 0 (P is not necessarily shown to scale)How far is P from the center of the cablea 200 mmb 240 mmc 245 mm
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
XX
XX X
X X10cm
coil
coilTop View
B B
side view[The two ends of the wire are connected (not shown) forming a closed circuit]
Bz
time
02 s
A single wire is wrapped into a coil with 200 turns and diameter 10 cm (see figures) The axis of the coil is aligned vertically The coil is placed in a uniform magnetic field of magnitude 20 T in the upward direction The direction of the field is suddenly reversed during a time interval of 02 s
Find the average emf in the coil during the reversala 0157 Vb 157 Vc 314 V
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Non-Simple Circuitsbull All circuits are governed by
loopnV 0 outin II
bull When capacitors ( V = QC) and inductors (V = L dIdt) are involved the currents are time-dependent
RI I
C
a
b
Cq
dtdqR
LRteR
I
LRteR
I 1 R
I I
a
b
L IRdtdIL
RCteCq 1
RCteCq
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
AC Circuits
Resonance Im max when XL = XC
bull LCR Circuit
LC
R
CL XX Z
R
ImR
ImXL
ImXC
m
RXX CL
tan 22CL XXRZ
ZXXR
I m
CL
mm
22
LX L
CX C
1
High ldquoQrdquo High VL VC over narrow freq range
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
t
I
TL
R
ε
I
A B
The current as a function of time is shown in the right-hand side figure Assume R and L are ideal and Rgt 0 Lgt0
Which of the following graphs best describes the voltage across the inductor VL(t) (=VB ndash VA)
t
VL
T
a
t
VL
T
e
t
VL
T
b
t
VL
T
c
t
VL
T
d
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
0
E BS
I
A
Side View
I
A
End-on View(current coming out
of the page)
In Lecture we discussed the Poynting vector as describing the direction of energy transport in electromagnetic waves The Poynting vector applies to other situations as well when there is a flow of electromagnetic energy Consider the following diagram showing a capacitor in the process of being charged up (ie a current is flowing onto the left plate and off of the right plate)
In the Side View what is the direction of S at the point A a down (ie radially inward)b up (ie radially outward)c into the paperd out of the papere undefined since |S|=0 in this situation
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
Circular Polarization
LCP = CCW
E
x
y
)sin(0 tkzEEx )cos(0 tkzEEy SLOW
E
TA
FASTEf
EsLCP
z4
I1
I2= I1
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
21 Light polarized along the x-axis is directed through a quarter waveplate (fast axis initially along x) and a polarizer (transmission axis also along x) as shown below [Assume the polarizer is ideal ie light polarized along the transmission axis is completely transmitted]
z
TAx
y
Eo
QuarterWaveplate
fast axis
As the fast axis of the quarter waveplate is rotated in the xy plane by a small angle the output intensity will decreasea Trueb False
