Lesson 7Lesson 7
Magnetic FieldsMagnetic FieldsMagnetic Force on moving Magnetic Force on moving chargechargeMotion of charge in Magnetic Motion of charge in Magnetic FieldFieldMagnetic Force on current Magnetic Force on current carrying conductorcarrying conductorTorque on current carrying Torque on current carrying looploopHall EffectHall EffectMagnetic DevicesMagnetic Devices
Some metallic objects cause other
metallic objects to accelerateThus must be producing a forceThis force is different to Gravitational
Force and Electric ForceCan plot the directions of lines force
#Lines of motion of effected objectsMAGNETIC FORCEMAGNETIC FORCE
Magnetic Force
Magnetic Field Lines
Properties of Magnetic FieldsProperties#Moving charge (current) is effected
Force proportional ³Q and v
FB = 0 if v parallel to Magnetic
Field BFB perpendicular to v and BFB on positive charge opposite to
that on negative charge FB proportional to Sin of the
angle v makes with B
Putting these properties together one sees
FB Qv B
compare this to
FE QE
Mathematical Expression
Vector Product
A Ax i Ay j Az k
B Bx i By j Bz k
A B i j k
Ax Ay Az
Bx By Bz
Ay Az
By Bzi
Ax Az
Bx Bzj
Ax Ay
Bx Byk
Ay Bz Az By i Ax Bz Az Bx j Ax By Ay Bx k
Vector Product I
A B BA
A A 0
A B 0 A parallel to B
A B (AB)Sin A B AB if A is perpendicular to B
A B is perpendicular to A and B
Vector Product II
i j k
i k j
jk i
Unit vector products
Work done by Magnetic Field�
Work Done by Magnetic Force Work Done by Magnetic Force on chargeon charge
dWB FB ds Qv B ds 0
As v B is perpendicular to v and B
and v is parallel to ds
Compare to Electric ForceCompare to Electric Force
dWE FE ds QE ds 0
unless E is perpendicular to path of motion
which is never the case if E causes the motion.
Work done by Electric Field
ImplicationsThus Kinetic Energy of charge is not changed by Thus Kinetic Energy of charge is not changed by FFBB
Potential Energy is not changedPotential Energy is not changed
K 1
2m v 2 0
v 0
i.e. speed is constant under influence of B
but v can change direction
FB Q v B
B FB Q v
N
Cm
s
N
AmT (Tesla)
SI units
Magnetic FluxMagnetic Flux B
For constant field over flat area perpendicular
with area vector A to magnetic field B
B B A
in general
B B d
surface A
B Tm2 Wb (Weber )
Right Hand Rule
FB Qv B Q vy j Bz k
Qv y Bz j k Qv y Bz
i j k
0 1 0
0 0 1
Qv y Bz i
kk
iijj
vv
FFBB
B
Magnetic Force on Moving Charge
FB t Qv t BMathematical expression
Uniform Circular Motion of Charge
Uniform Circular Motion
i
j k
Mathematical analysis
ac v2
r F
mQvBm
mv2
rQvB
r mvQB
radius of motion
v = r
=BQm
angular speed =angular freqency ( rads )
f
2 BQ
2 m frequency (cps = Hz )
T 1f2 mBQ
period of the motion .
Magnetic BottlesIf initial v has a component in k direction
v t 1
mFBt v 0
vx t i vy t j vz 0 k
magnetic bottles
Vector Integral I
a
b
rb
ra
I
ds
B
vector integral
Vector Integral II
a
brb
ra
I
ds
Brb- ra
dsSum
Vector Integral III
Magnetic Force on Current
Force on current carrying conductor in
a uniform magnetic field :
FB I d s B
a
b
Il B
where l ra rb
dQ v dQ ds
dtˆ s dQ
dtds
s Ids s Id s
Magnetic Force on Current Loop I
I
Force on Current Loop
B
k
ji
Magnetic Force on Current Loop II
FB IlB0
l r a ra 0
Total Force on Loop
Torque on Current Loop I
I
y
z
B
Torque on Current Loop
k
ji
Torque
r F
Torque on Current Loop II
Torque on Current Loop IIIForce on segment 1
F1 IzBk j IzBi
Force on segment 3
F3 IzBk j IzBi
Forces on segment 2 and 4 are 0 as
magnetic field and wire are parallel
F Iyjj 0
Torque about central axis
= 1 2 IzB y k IzB k IzBy k IABk
A Area of loop
Magnetic Moment
Torque on Current Loop in yz plane in
a uniform magnetic field in positive y direction
= IABk
In general the torque on a current loop in magnetic field is
= M B
M IA, where A is the area vector of the loop,
with the orientation given by the current flow.
Oriented Area Vector
I
k
ji
A=Ai: Area vector with orientation
A
Hall EffectHall EffectHall Effect II
BB
vvdd
FFEE FFBB
EE
d
+ -
AA
t
V
Mathematical analysisAt Equlibrium
F E F B
E V
dF E eE F B vdeB
vd E
B
Remember vd J
ne I
neA
IneA
EB
n BIEAe
BIV
ddte
BIVte
can measure vd by measuring
the PD V.
Measurement
Lorentz Force
F FE FB QE Qv B
Lorentz Force