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Chapter 28 and 29Hour 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 ChargeAn electric charge produces an electric field:
rˆ
r 2
P15-3
E 1 q
rˆ
4 o
rˆ : unit vector directed from q to Pe0 = 8.85 1012 C2 / Nm2 permittivity of free space
Magnetic Field Of Moving Charge
Moving charge with velocity v produces magnetic field:
r 2
q v x
rˆ4oB
ˆr:
r
ˆ
P
unit vector directed from q to P
P15-4
permeability of free space 4 107 T m/A0
The Biot-Savart LawCurrent element of length ds carrying current I produces a magnetic field:
r 2
0 I ds
rˆ4dB
http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/03-CurrentElement3d/03-cElement320.html
P15-5
The Right-Hand Rule #2
zˆ ρˆ
φˆ
P15-6
Animation: Field Generated by a Moving Charge
(http://ocw.mit.edu/ans7870/8/8.02T/f04/visualizations/magnetostatics/01-MovingChargePosMag/01- MovChrgMagPos_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
II
I
Find the magnetic field B at the center (P)
P
P15-9
Example : Coil of Radius R
Consider a coil with radius R and current I
I I
IP
1) Think about it:• Legs contribute nothing
I parallel to r• Ring creates B field into pageB field contributions from all
segments are in the same direction.
2) Choose a ds3) Pick your coordinates4) Write Biot-Savart
P15-10
Example : Coil of Radius RIn the circular part of the coil…
d s ˆr | d s ˆr | ds
Ir
ˆds
II
0
4 r2 I ds
rˆ
dB Biot-Savart:
0
4 r2
I ds
0 I R d
R24
0 I d
P15-11
4 R
B field contributions from all segments are in the same direction: into the screen.
Example : Coil of Radius RConsider a coil with radius R and current I
dB 0 I d4 R
2 I dB dB 0 0 4 R
2
0
0 I 0 24 R 4 R
Id
I I
I
ds
0
P15-12
into page2R
I B
zˆ
Example : Coil of Radius R
Notes:•This is an EASY Biot-Savart problem:
• No vectors involved•This is what I would expect on exam
II
P I
into page2R
0 I B
P15-13
PRS Questions:B fields Generated by Currents
P15-14
B Field from Coil of Radius Rat location P along its axis
Consider a coil with radius R and carrying a current I
This is much harder than what we just did!
P15-15
What is B at point P?
P15-16
Think about it: 1) Choose a ds is along thedirection of dBdB’s y component canceldue to symmetry. dB’s x component adds up2) Pick your coordinates
ds rˆ
What is B at point P?
3) Write Biot-Savart
dBx = dB cosq cos q = R/(x2 + R2)1/2
4) Integrate dB
2
0 03/2 3/22 2 2 22 2
x
I R mB
x R x R
2
0 03/2 3/22 2 2 22 2
I R
x R x R
���������������������������� mB x
2( )m I Area I R 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 Iproduces a magnetic field:
r 2
P18-18
0 I ds
rˆ4dB
If the wire length is infinite, q1=0 ; q2= p=B m0 /(2I pa)
If the wire is half infinity, q1=0 ; q2= /2p=B m0 /(4I pa)