Lecture 15-Lecture 15-11 What we learned from last class

• Define magnetic dipole moment by

IAn

B

where n is normal to the loop with RHR along I.

BU

Hall effect:

Determines the sign and number of carriers.

Measures B.

Lecture 15-Lecture 15-22

Van Allen belts

Lecture 15-Lecture 15-33Sources of Magnetic Fields

024

μ q v rdBπ r

70 2

N4 10A

Permeability constant

• Moving point charge:

024

μ I d l rdBπ r

• Bits of current:

I

Biot-Savart Law

The magnetic field “circulates” around the wire.

Lecture 15-Lecture 15-44Magnetic field due to a current loop

Principle of superposition:

024

Id l rBr

At the center, sin ( / 2)dl r dl x dl x

02

0

24I x I xR

BR

dl

On axis generally,

0 02 2 24 4

I dl I dldBr R x

0

3/ 22 2sin

4xI RdB dB dl

R x

30

2 2 3/ 22 ( )x xI RB dB

R R x

xdB

Lecture 15-Lecture 15-55Circular Loop Current as a Magnetic Dipole

Lecture 15-Lecture 15-66Magnetic Field of Circular Arc Current

0 02 2

sin 904 4

ids idsdBR R

0 0

2 2 2 2

dsiR R

iBR

I runs clockwise in the closed loop wires below:

What is B at center?

Lecture 15-Lecture 15-77 PHYS241 – warm-up

A circular current loop lies in the plane perpendicular to this sheet with its axis along the x-direction, and produces magnetic field B as shown. What is the direction of the current at the top end of this loop?

a. Out of the sheetb. Into the sheetc. Along +x axisd. Along x axise. Current is zero.

x

Lecture 15-Lecture 15-88Gauss’s Law for Magnetism

sources

No sources

Gauss’s Law

Gauss’s Law for Magnetism

s

sE

qadE0

sB adB 0

Lecture 15-Lecture 15-99Magnetic field of a solenoid

• A constant magnetic field could be produced by an infinite sheet of current. In practice, however, it is easier and more convenient to use a solenoid.

• A solenoid is defined by a current I flowing through a wire that is wrapped n turns per unit length on a cylinder of radius R and length L.

L

R

Lecture 15-Lecture 15-1010Magnetic field of a solenoid (continued)

20

2 2 3/ 2

24 ( )x

I RdB n dxR x

Contribution to B at origin from length dx

one turn # turns in length dx

2

1

202 2 3/ 2

2 10 2 2 2 2

2 1

24 ( )

12

x

x x

dxB R nIR x

x xnIx R x R

1 2,As x x

0xB nI(or for L>>R)

(half at ends)

Lecture 15-Lecture 15-1111Solenoid’s B field synopsis

// to axis

• Long solenoid (R<<L):

B inside solenoid

B outside solenoid nearly zero

(not very close to the ends or wires)

Lecture 15-Lecture 15-1212 RHIC STAR Experiment

STAR

Lecture 15-Lecture 15-1313Magnetic field of a Straight Current

I

2

1

0 cos4μ IB dB d

π R

y

0 02 2sin cos

4 4μ I dx μ I dxdB

π r π r

tan ,x R θ2 2

2sec r rdx R dθ R d dR R

02 1sin sin

4IBR

1 2/ 2, / 2

Infinite straight current

00 224

IR

IBR

http://falstad.com/vector3dm/

Lecture 15-Lecture 15-1414Two Perpendicular Currents

I1

I2

FB 1,2

FB 1,3

I3

B

B

FB2,1

Lecture 15-Lecture 15-1515 PHYS241 - Quiz A

A circular current loop lies on the xy-plane as shown,where the current is clockwise as seen from the positive z-axis. What is the direction of the B field at point A?

a. Along +x axisb. Along +y axisc. Along +z axisd. Along z axise. Along x axis

z

x

y

A

I

Lecture 15-Lecture 15-1616 PHYS241 - Quiz B

A circular current loop lies on the xy-plane as shown, where the current is clockwise as seen from the positive z-axis. What is the direction of the B field at point A?

a. Along +x axisb. Along +y axisc. Along +z axisd. Along z axise. Along x axis

z

x

y

A

I

Lecture 15-Lecture 15-1717 PHYS241 - Quiz C

A circular current loop lies on the xy-plane as shown, where the current is clockwise as seen from the positive z-axis. What is the direction of the B field at point A?

a. Along +x axisb. Along +y axisc. Along +z axisd. Along z axise. Along x axis

z

x

y

A I