Electricity and Magnetism
Review 3: Units 12-16
Mechanics Review 2 , Slide 1
Review Formulas

 
Magnetic Force on moving charges F  qv  B
Magnetic Force on current

 
segments in uniform magnetic F  IL  B
field
  
Torque on a current loop in     B
uniform magnetic field
 0 I d s  rˆ Magnetic field from
Biot-Savart Law dB 
infinite line of current
4 r 2
 
Ampere’s Law
Can be used to determine B field in  B  d   o I enc
0 I
B
2 r
symmetrical cases
Motional emf on a conductor of
length L in a uniform magnetic field
FB
emf  EL  vBL
+
FE
+
E
-
v
Example
y
Two parallel horizontal wires are
located in the vertical (x,y) plane as
shown. Each wire carries a current
of I  1A flowing in the directions
shown.
What is the B field at point P?
y
I1  1A
.
4cm
3cm
x
4cm
z
P
I2  1A
Side view
Front view
What is the direction of B at P produced by the top current I1?
What is the direction of B at P produced by the bottom current I2?
What is the direction of B at P?
y
y
y
.
P
z
z
.
.
90o
z
P
P
Electricity & Magnetism Unit 14, Slide 3
Example
y
Two parallel horizontal wires are
located in the vertical (x,y) plane as
shown. Each wire carries a current
of I  1A flowing in the directions
shown.
What is the B field at point P?
y
I1  1A
.
4cm
x
4cm
3cm
z
P
I2  1A
Front view
Side view
What is the magnitude of B at P produced by the top current I1?
m0 I
B=
=
2pr
(
)
4p ´10-7 ´1
2pr
y
-6
= 4.0 ´10 T
.
4cm
z
r
r  32 + 42  5cm
3cm
Electricity & Magnetism Unit 14, Slide 4
Example
y
Two parallel horizontal wires are
located in the vertical (x,y) plane as
shown. Each wire carries a current of
I  1A flowing in the directions
shown.
What is the B field at point P?
y
I1  1A
.
4cm
x
4cm
3cm
z
P
I2  1A
Front view
Bbot = Btop  4 x 10-6 T
y
.
4cm q
z
3cm
q
B1
5cm
q
q
B2
B1x  B1 cos q
B2 x  B2 cosq
æ4ö
Bx = 2B1 cosq = 2 ´ 4 ´10-6 ´ ç ÷ = 6.4 ´10-6 T
è 5ø
Example
A particle of charge q and mass m is
accelerated from rest by an electric field E
through a distance d and enters and exits a
region containing a constant magnetic field
B. Assume q,m,E,d, and x0 are known.
What is B?
x0/2
q,m
E
d
enters here
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
What is v0, the speed of the particle as it enters the magnetic field ?
Conservation of Energy
Initial: Energy  U  qV  qEd
Final: Energy  KE  ½ mv02
1
2
mv  qEd
2
o
2qEd
vo 
m
Example
A particle of charge q and mass m is
accelerated from rest by an electric field E
through a distance d and enters and exits a
region containing a constant magnetic field
B. Assume q,m,E,d, and x0 are known.
What is B?
x0/2
q,m
d
enters here
1
What is the radius of path of particle? R  xo
2
R xo/2
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
XXXXXXXXX
E
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
Example
A particle of charge q and mass m is
accelerated from rest by an electric field E
through a distance d and enters and exits a
region containing a constant magnetic field
B. Assume q,m,E,d, and x0 are known.
What is B?
x0/2
q,m
E
d
enters here
exits here
XXXXXXXXX
X X X X X X X X X x0
XXXXXXXXX
XXXXXXXXX
B
B
vo 
2qEd
m


F  ma
R  12 x0
vo2
qvo B  m
R
m 2
B
q xo
2qEd
m
m vo
B
q R
2
B
xo
2mEd
q
Example
A rectangular loop (h  0.3m L  1.2 m) with total
resistance of 5W is moving away from a long
straight wire carrying total current 8 amps. What is I
the induced current in the loop when it is a
distance x  0.7 m from the wire?
B into page
x
What is the emf induced on the left segment?
(top point is positive)    o I hv
left
2x
What is the emf induced on the top and
bottom segments?
etop = ebot = 0
o I
B
2x
L
B into page
I
v
h
x
 
vB
v
h
L
Example
A rectangular loop (h  0.3m L  1.2 m) with total
resistance of 5W is moving away from a long
straight wire carrying total current 8 amps. What is
I
the induced current in the loop when it is a
distance x  0.7 m from the wire?
B into page
x
What is the emf induced on the right segment?
(top point is positive)   o I hv
right
o I
o I
hv hv
2x
2 ( L + x)
 left >  right
Clockwise current
L
o I
B
2x
2 ( L + x)
What is the total emf induced in the loop?
 loop 
v
h
I loop 
 loop
R
 I 1
1 
I loop  o hv 
2R  x L + x 
Example
A rectangular loop (h  0.3m L  1.2 m) with
total resistance of 5W is moving away from a
long straight wire carrying total current 8 amps.
B into page
I
h
I
x
L
What is the direction of the force exerted
by the magnetic field on the loop?
B into page
I
Total force from B
Points to the left
v
Example
The loop has a radius of R and carries a steady current of I.
Find the field at point P.
Bx =
(
mo I a 2
2 a +x
2
2
)
3
2
Example
A conducting bar (green) rests on two
frictionless wires connected by a resistor as
shown and is placed in a uniform magnetic
field into the screen.
Equivalent circuit
R
I
V
Bar
Opposite forces on charges
Charge separation
E  v0B
emf  EL  v0BL
I = emf/R
XXXXXXXXX
F
B
+-
F
XXXXXXXXX
Fb = qv0B
v0

 
F  IL  B
 vBL 
F 
 LB
R


 vBL 
2
P  Fv  
 LBv  I R
 R 
Example
A rectangular loop of sides L and w rotates with constant
angular speed ω in a region containing a constant magnetic
field B as shown. The side view of the loop is shown at a
particular time t during the rotation. What is the induced
emf on the loop at that time?
Example
Two infinitely long wires are lying on the ground a distance
a apart. A third wire of length L and mass M carries a
current I1 and is levitated above them as shown.
What current I2 must the infinitely long wires carry so that
the three wires form an equilateral triangle?
Example
Find the magnetic field at point O due to the wire segment
in the picture.
mo I
B=
q
4p a
q will be in radians
For a full circle: θ = 2π
B=
mo I
m I
m I
q = o 2p = o
4p a
4p a
2a
Example
y
An infinitely long cylindrical shell with inner
radius a and outer radius b carries a uniformly
distributed current I out of the screen.
Find |B| as a function of r.
I
r
a
What is |B| for r < a?
b
x

so B  0
What is |B| for r > b?
dl
r
I
B
a
b
o I
B
2r
Electricity & Magnetism Unit 15, Slide 17
Example
y
What is |B| for a < r < b ?
I
r
What is the current density j
(Amp/m2) in the conductor?
a
j = I / area
area   b2 -  a 2
I
j 2
 b -  a2
 
B  2r  o jAenc
 B  d   oIenc
I
2
2
B  2r   o


(
r
a
)
2
2
 (b - a )
o I ( r 2 - a 2 )
B
 2 2
2r (b - a )
b
x
Example
Find the field at a point at distance r from the center
of the toroid. The toroid has N turns of wire.
 
B

d



I
o
enc

I enc = NI
mo NI
B=
2p r
Example
Find the torque exerted on a rectangular loop of sides w and h and
carrying a current I in a uniform magnetic field B.

  B


I
For a rectangular
loop of sides w and h

m = IA = Iwh
B
In this case  is out of the page (using right hand rule)
z

x

  B


B
y
is up (turns  toward B)
t = IwhBsin(q )