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)