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: rˆrˆ r 2r 2 P E  1 q r ˆ 4  o  r ˆ : unit vector directed from q to P    10  12 C 2 /  m 2 permittivity of free space

Magnetic Field Of Moving Charge Moving charge with velocity v produces magnetic field: r 2r 2 q v x rˆq v x rˆ 44 o B B    ˆr:ˆr: rˆrˆ P unit vector directed from q to P P permeability of free space   4   10  7 T  m/A 0

The Biot-Savart Law Current element of length ds carrying current I produces a magnetic field: r 2r 2 0 I ds  rˆI ds  rˆ 44 dB dB   P15 - 5

The Right-Hand Rule #2 zˆ  ρˆ  φˆzˆ  ρˆ  φˆ P15 - 6

Animation: Field Generated by a Moving Charge ( MovChrgMagPos_f223_320.html) 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 I 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 I P 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 P

Example : Coil of Radius R In the circular part of the coil… d s  ˆr| d s  ˆr | ds Id s  ˆr| d s  ˆr | ds I rˆrˆ dsds  I I 04r04r 2  Ids  rˆ Ids  rˆ dB dB  Biot-Savart: 04r04r 2  Ids   0 IR d 0 IR d R2R2 44  0 Id 0 Id P R4R B field contributions from all segments are in the same direction: into the screen.

Example : Coil of Radius R Consider a coil with radius R and current I dB  0 IddB  0 Id 4R4R 2Id2Id B   dB  0B   dB  0 0 4R4R  2020 0 I0 I 0 22 4 R4 R4 R4 R II d d  I I I  dsds  0 P into page 2R2R II B B  zˆzˆ

Example : Coil of Radius R Notes: This is an EASY Biot-Savart problem: No vectors involved This is what I would expect on exam I I P I into page 2R2R 0 II B B  P

PRS Questions: B fields Generated by Currents P

BFieldfromCoilofRadiusR at 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! P What is B at point P?

P Think about it: 1)Choose a ds is along the direction of dB dB’s y component cancel due to symmetry. dB’s x component adds up 2) Pick your coordinates ds  rˆds  rˆ What is B at point P? 3) Write Biot-Savart dB x = dB cos  cos  = R/(x 2 + R 2 ) 1/2 4) Integrate dB

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: r 2r 2 P I ds  rˆI ds  rˆ 44 dB dB 

If the wire length is infinite,         a   f the wire is half infinity,         a 