Chapter 28 and 29
Hour 1: General introduction, Gauss’ Law
Magnetic force (28.1) Cross product of vectors.
Hour 2: Currents create B Fields: Biot-Savart,
B field of loops (magnetic moment). (28.2)
Hour 3: Use Ampere’s Law to calculate B fields (28.3)
Hour 4: Charged particle’s motion in B field. (29.1)
Hour 5: B field force & torque on wires with I (29.2)
Hour 6: Magnetic materials (29.4)
Sources of Magnetic Fields:
Moving charges (current)
The Biot-Savart Law
P15-2
Electric Field Of Point Charge
An electric charge produces an electric field:
r̂
1 q
E  4 r 2 r̂
o
r̂
: unit vector directed from q to P
  8.85 10 12 C2 / Nm2 permittivity of free space
P15-3
Magnetic Field Of Moving Charge
Moving charge with velocity v produces magnetic field:
P
r̂
0  4 107 T  m/A
o q v x r̂
B
4 r 2
r̂: unit vector directed
from q to P
permeability of free space
P15-4
The Biot-Savart Law
Current element of length ds carrying current I
produces a magnetic field:
0 I ds  r̂
dB 
2
4 r
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.html
P15-5
The Right-Hand Rule #2
ẑ ρ̂  φ̂
P15-6
Animation: Field Generated by a
Moving Charge
(http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/01-MovingChargePosMag/01MovChrgMagPos_f223_320.html)
P15-7
Demonstration:
Field Generated by Wire
Clicker questions
P15-8
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
Find the magnetic field B at the center (P)
P15-9
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I
P
I
1) Think about it:
• Legs contribute nothing
I parallel to r
• Ring creates B field into page
B field contributions from all
segments are in the same
direction.
2) Choose a ds
3) Pick your coordinates
4) Write Biot-Savart
P15-10
Example : Coil of Radius R
In the circular part of the coil…
d s  r̂  | d s  r̂ | ds
I
Biot-Savart:
0 I
dB 
4
r̂

0I

4
ds
B field contributions from all

0I

segments are in the same
4
direction: into the screen.
I
I
ds  r̂ 0 I ds

2
2
4

r
r
R d
R2
d
R
P15-11
Example : Coil of Radius R
Consider a coil with radius R and current I
I
I

ds
I

0 I d
dB 
4 R
0 I d
B   dB  
0 4 R
2


0I
0I

d 
2 


4 R 0
4 R
2
B
0 I
ẑ
2R
into page
P15-12
Example : Coil of Radius R
I
I
B
P
I
0 I
2R
into page
Notes:
•This is an EASY Biot-Savart problem:
• No vectors involved
•This is what I would expect on exam
P15-13
PRS Questions:
B fields Generated by Currents
P15-14
B Field from Coil of Radius R
at location P along its axis
Consider a coil with radius R and carrying a current I
What is B at point P?
This is much
harder than
what we just
did!
P15-15
What is B at point P? Think about it:
1) Choose a ds
ds  r̂ is along the
direction of dB
dB’s y component cancel
due to symmetry.
dB’s x component adds up
2) Pick your coordinates
3) Write Biot-Savart
dBx = dB cos
cos  = R/(x2 + R2)1/2
4) Integrate dB
Bx 
0 I  R 2

2 x  R
2
2

3/2


0 m
2 x  R
2
2

3/2
P15-16
B
0 I  R

2
2 x  R
2
2

3/2
x
m  I ( Area)  I  R

0 m
2 x  R
2
2

3/2
2
A current loop with area A and carrying current I has
a magnetic dipole moment m.
m = I A.
The magnetic dipole moment is a vector,
whose direction is perpendicular to the loop.
Right hand rule:
Curve four fingers
along the current’s
direction and the
thumb points to ’s
direction.
The Biot-Savart Law
Current element of length ds carrying current I
produces a magnetic field:
0 I ds  r̂
dB 
4 r 2
P18-18
If the wire length is infinite, 10 ; 2 
B0I/2a
If the wire is half infinity, 10 ; 2 /2
B0I/4a