Physics 202, Lecture 13

Today’s Topics

Magnetic Forces: Hall Effect (Ch. 27.8)

Sources of the Magnetic Field (Ch. 28) B field of infinite wire

Force between parallel wires

Biot-Savart Law Examples: ring, straight wire

Ampere’s Law: infinite wire, solenoid, toroid

The Hall Effect (1)

positive charges moving counterclockwise: upward force, upper plate at higher potential

negative charges moving clockwise: upward forceUpper plate at lower potential

Equilibrium between electrostatic & magnetic forces:

Potential difference on current-carrying conductor in B field:

F

up=qvdB Fdown =qEind =q

VH

wVH =vdBw="Hall Voltage"

The Hall Effect (2)

VH =vdBw=IBnqt

I =nqvdA=nqvdwt

RH ≡VH

IB=

1nqtHall coefficient:

Hall effect: determine sign,density of charge carriers

(first evidence that electrons are charge carriers in most metals)

Text: 27.47

Magnetic Fields of Current Distributions

Two ways to calculate E directly:

"Brute force"– Coulomb’s Law

"High symmetry"– Gauss’ Law

d

rE =

14πε0

dqr2

r̂

rE gd

rA∫ =

qin

ε0ε0 = 8.85x10-12 C2/(N m2): permittivity of free space

rF =

kq ′qr2

r̂

Recall electrostatics: Coulomb’s Law)

rF =q

rE (point charges)

ε0 =(4πk)−1

Magnetic Fields of Current Distributions

I

– Biot-Savart Law (“Brute force”)

–AMPERIAN LOOP

2 ways to calculate B:

– Ampere’s Law (“high symmetry”)

d

rB =

μ0 I4π

drl ×r̂r2

rBgd

rl—∫ =μ0 Ienclosed

μ0 = 4πx10-7 Tm/A: permeability of free space

rF = Id

rl ×

rB∫

(2 circuits)

rF =km Id

rl × ′I d

r′l ×

r̂r2∫∫

μ0 =4πkm

Magnetic Field of Infinite Wire

rB =

μ0 I2πr

θ̂

closed circular loops centered on current

Result (we’ll derive this both ways):

right-hand rule

Which figure represents the B field generated by the current I?

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x x x x x x x x x x x x x x x x x x x x x x x x x x x x x x

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Quick Question 1: Magnetic Field and Current

Quick Question 2: Magnetic Field and Current

Which figure represents the B field generated by the current I?

Forces between Current-Carrying WiresA current-carrying wire experiences a force in an external

B-field, but also produces a B-field of its own.

dIa

Ib

rF

rF

dIa

Ib

rF

rF

Parallel currents: attract

Opposite currents: repel

Magnetic Force:

Example: Two Parallel Currents

Force on length L of wire b due to field of wire a:

Ba =μ0 Ia

2πd Fb = Ibd

rl ×

rBa =∫

μ0 IbIaL2πd

dIa

Ib

rF

rF

xForce on length L of wire a due to field of wire b:

Bb =μ0 Ib

2πd Fa = Iad

rl ×

rBb =∫

μ0 IaIbL2πd

Text: 28.4, 28.58

Biot-Savart Law...

I B field “circulates” around the wire

Use right-hand rule: thumb along I, fingers curl in direction of B.

X

dB

dl

rθ

…add up the pieces

d

rB =

μ0 I4π

drl ×r̂r2

=μ0 I4π

drl ×

rr

r3

B Field of Straight Wire, length L

(Text example 28-13) B field at point P is:

When L=infinity (text example 28-11):

)cos(cos4 21

0 θθπμ

−=aI

B

θ1 θ2

a

IB

πμ2

0=

Text: 28.43

(return to this later with Ampere’s Law)

B Field of Circular Current Loop on Axis

Use Biot-Savart Law (text example 28-12)

2/322

20

)(2 Rx

IRBx +

=μ

Bcenter =μ0 I2R

See also: center of arc (text example 28-13)

Text: 28.37

B of Circular Current Loop: Field Lines

B

Ampere’s LawAmpere’s Law:

applies to any closed path, any static B field useful for practical purposes

only for situations with high symmetry

Generalized form (valid for time-dependent fields): Ampere-Maxwell (later in course)

rBgd

rl =μ0 Iencl

anyclosed path

—∫

I

dl

Use symmetry and Ampere’s Law:

Ampere’s Law: B-field of Straight Wire

I

vBgd

rl =μ0—∫ Iencl

vBgd

rl = Brdθ =2πrB=—∫ μ0 Iencl—∫ =μ0 I

B =

μ0 I2πr

Quicker and easier method for B field of infinite wire (due to symmetry -- compare Gauss’ Law)

Choose loop to be circle of radius R centered on the wire in a plane to wire.

Why?Magnitude of B is constant (function of R only)Direction of B is parallel to the path.

B Field Inside a Long Wire

Total current I flows through wire of radius a into the screen as shown.

x x x x x

x x x x x x x x

x x x x x x x x x

x x x x x x x x

x x x x x a

r

What is the B field inside the wire?

By symmetry -- take the path

to be a circle of radius r:

rBgd

rl =B2πr—∫

•Current passing through circle: Iencl =r2

a2 I

rBgd

rl =B2πr =μoIencl—∫ Bin =

μ0 Ir2πa2

Text example: 28-6

B Field of a Long Wire

• Outside the wire: ( r > a )

B =

μ0 I2π r

Inside the wire: (r < a)

B =

μ0 I2π

ra2

r

B

a

Text: 28.31

Ampere’s Law: Toroid

(Text example 28-10)

Bin =μ0NI2πr

N = # of turns

Show:

Ampere’s Law: Toroid

Toroid: N turns with current I.

Bθ=0 outside toroid!

Bθ inside: consider circle of radius r, centered at the center of the toroid.

x

x

x

x

x

x

x

x

x x

x

x x

x

x

x

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r

B

(Consider integrating B on circle outside toroid: net current zero)

rBgd

rs =B2πr =μoIenclosed—∫

B =μ0NI2πr

B Field of a Solenoid

Inside a solenoid: source of uniform B field

If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude.

• Solenoid: current I flows through a wire wrapped n turns per unit length on acylinder of radius a and length L.

L

aTo calculate the B-field, use Biot-Savart

and add up the field from the different loops.

In this limit, can calculate the field using Ampere's Law!

Ampere’s Law: Solenoid

The B field inside an ideal solenoid is: nIB 0μ=

ideal solenoid

n=N/L

segment 3 at

Right-hand rule: direction of field lines

Solenoid: Field Lines

Like a bar magnet! (except it can be turned on and off)

Text: 28.68

north pole south pole