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The Magnetic Field
The force on a charge q moving with a velocity F
v
)( BvEqF
The magnitude of the force
sinqvBF
sec)//(][ meterCoulombNewtonsB
gausssmCNewtonmwteslaT 42 10//1/1)(1
TGaussBEarth4101
left-hand ruleright-hand rule
Motion in magnetic field
1) Uniform , B
Bv
2) Uniform , B
Bv
3) Nonuniform B
0E
BvqF
qvBFBv
amF
0 FmaF rr
r
vmmrqvB
22 ri
qB
mvr
The angular velocity
m
qB
qBmvv
r
v
0|| FBv
The angular velocity (cyclotron frequency )
m
qB
qBmvv
r
v
2
f
does not depend on velocity!
The force is always perpendicular to velocity, so it cannot change the magnitude of the velocity, only its direction. The work done by the magnetic force is zero! Motion of a charged particle under the action of a magnetic field alone is always motion with constant speed.
Electron motion in a microwave oven
A magnetron in a microwave oven emits electromagnetic waves with frequency f=2450 MHz. What magnetic field strength is required for electrons to move in circular paths with this frequency?
Using Crossed and Fields E
B
Velocity selector
0 qEqvB
vBE
B
Ev independent of the mass of the particle!
Mass spectrometer
qB
vmR 1
1
B
Ev
21
1 qB
EmR
22
2 qB
EmR
Thomson’s e/m experiment
1897: Cavendish Laboratoryin Cambridge, England
m
eVveVmv
2
2
1 2
B
Ev
2
2
2
2
VB
E
m
e
m
eV
B
E
Current carrying wires
1820 Hans Christian Oersted
Hans Christian Ørsted
Gauss’s Law
The total flux of electric field out of any closed surface is equal to the charge contained inside the surface divided by .0
S
enclosedQSdE
0
Conductors and insulators
Charges reside at the surface of the conductor
Conductor
E=0+
+ ++
+
+++
++
+++
+
+
Electric field of a ball of charge
30
20
4
1
4
1
R
rQERr
r
QERr
Q
Electric field outside of a charged sphere is exactly the same as the electric field produced by a point charge, located at the center of the sphere, with charge equal to the total charge on the sphere.
Insulating sphere with charge Q uniformly spread throughout the volume
A
E
r
204
1
r
QEAr
rA
QEAr
304
A
Conducting sphere with charge Q
A
E
r
204
1
r
QEAr
0 EAr
V
r
A
A
Q
04
1
r
QVAr
04
1
A
QVAr
04
1
A Charged, Thin Sheet of Insulating Material
++
+
+
++
+
++
++
02
E
A field in a cavity of a conductorFaraday’s cage
Electric field near a surface of a conductor
al
cap
EaEdSSdE
0a
Ea 0
E
A
QLL
A
ALbottomVtopV
000
)]()([
0AQL
V
The capacitance is:
L
A
AQLQ
V
QC 0
0
Spherical capacitor; Cylindrical capacitor
Capacitors in series: ...1111
321
CCCCtot
Capacitors in parallel: ...321 CCCCtot
22
2
1
2
1Q
CCVW
faradC ][
If the capacitors were initially uncharged,
Current Density
S
Sdji
Consider current flowing in a homogeneous wire with cross sectional area A.
jAdSjjdSSdjiA A A
A
ij for j =Const only!
Current, Ohm’s Law, Etc.
dt
dQi
)(;:' VoftindependenConstRi
VRLawsOhm
A
lR
Ej
jE
For steady state situation
1.Kirchhoff’s junction rule: The algebraic sum of the currents into any junction is zero.
2.Kirchhoff’s loop rule: The algebraic sum of the potential differences in any loop must be zero.
0 Sdj
0 rdE
Joule’s Law
R
VRiViP
22
Resistors in parallel:
Resistors in series:
21 RRR
21
111
RRR