Physics 1202: Lecture 12Today’s Agenda

• Announcements:– Lectures posted on:

www.phys.uconn.edu/~rcote/– HW assignments, solutions etc.

• Homework #4:Homework #4:– Not this week ! (time to prepare midterm)Not this week ! (time to prepare midterm)

• Midterm 1:– Friday Oct. 2– Chaps. 15, 16 & 17.

Magnetic Force on a Current

or

Current loop & Magnetic Dipole Moment

• We can define the magnetic dipole moment of a current loop as follows:

direction: right-hand rule

• Torque on loop can then be rewritten as:

• Note: if loop consists of N turns, = N A I

magnitude: A I

A I Bsin

B

x

.FF w

• If plane of loop is not to field, there will be a non-zero torque on the loop!

• No net force

Calculation of Magnetic Field

• Two ways to calculate the Magnetic Field:• Biot-Savart Law:

• Ampere's Law

• These are the analogous equations for the Magnetic Field!

"Brute force" I

"High symmetry"

0= 4X 10-7 T m /A: permeability (vacuum)

Magnetic Field of Straight Wire

Direction of B:right-hand rule

Lecture 12, ACT 1• I have two wires, labeled 1 and 2, carrying equal

current, into the page. We know that wire 1 produces a magnetic field, and that wire 2 has moving charges. What is the force on wire 2 from wire 1 ?

(a) Force to the right (b) Force to the left (c) Force = 0

Wire 1

IX

Wire 2

IX

Force between two conductors

• Force on wire 2 due to B at wire 1:

• Total force between wires 1 and 2:

• Force on wire 2 due to B at wire 1:

• Direction:attractive for I1, I2 same directionrepulsive for I1, I2 opposite direction

Circular Loop

x

•

z

R

R

• Circular loop of radius R carries current i. Calculate B along the axis of the loop:

r B

r

z

B

• Symmetry B in z-direction.

>

>I

• At the center (z=0):

• Note the form the field takes for z>>R: for N coils

Lecture 12, ACT 2• Equal currents I flow in identical

circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.– What is the magnetic field Bz(A)

at point A, the midpoint between the two loops?

(a) Bz(A) < 0 (b) Bz(A) = 0 (c) Bz(A) > 0

Lecture 12, ACT 2• Equal currents I flow in identical

circular loops as shown in the diagram. The loop on the right (left) carries current in the ccw (cw) direction as seen looking along the +z direction.

(a) Bz(B) < 0 (b) Bz(B) = 0 (c) Bz(B) > 0

– What is the magnetic field Bz(B) at point B, just to the right of the right loop?

B Field of a Solenoid

• A constant magnetic field can (in principle) be produced by an sheet of current. In practice, however, a constant magnetic field is often produced by a solenoid.

• If a << L, the B field is to first order contained within the solenoid, in the axial direction, and of constant magnitude. In this limit, we can calculate the field using Ampere's Law.

L• A solenoid is defined by a current I flowing

through a wire which is wrapped n turns per unit length on a cylinder of radius a and length L.

a

B Field of a Solenoid

• To calculate the B field of the solenoid using Ampere's Law, we need to justify the claim that the B field is 0 outside the solenoid.

• To do this, view the solenoid from the side as 2 current sheets.

xxx xx

•• • ••• The fields are in the same direction in the region between the sheets (inside the solenoid) and cancel outside the sheets (outside the solenoid).

(n: number ofturns per unitlength)

Toroid• Toroid defined by N total turns

with current i.

• B=0 outside toroid!

• B inside the toroid.

x

x

x

xx

xx

x

x x

xx x

x

x

x

•

••

•

• •

•

••

•

• •

•

•

••

r

B

Magnetism in Matter• When a substance is placed in an external magnetic field Bo,

the total magnetic field B is a combination of Bo and field due to magnetic moments (Magnetization; M):

– B = Bo + oM = o (H +M) = o (H + H) = o (1+) H» where H is magnetic field strength

is magnetic susceptibility

• Alternatively, total magnetic field B can be expressed as:– B = m H

» where m is magnetic permeability» m = o (1 + )

• All the matter can be classified in terms of their response to applied magnetic field:

– Paramagnets m > o

– Diamagnets m < o

– Ferromagnets m >>> o

Faraday's Law

vB

N S

vB

S N

nB B

Induction Effects

v

vS N

vN S

N S

S N

• Bar magnet moves through coil

Current induced in coil

• Change pole that enters

Induced current changes sign

• Bar magnet stationary inside coil

No current induced in coil

• Coil moves past fixed bar magnet

Current induced in coil

Faraday's Law• Define the flux of the magnetic field B through a surface

A=An from:

• Faraday's Law:The emf induced around a closed circuit is determined by the time rate of change of the magnetic flux through that circuit.

The minus sign indicates direction of induced current (given by Lenz's Law).

nB B

Faraday’’s law for many loops

• Circuit consists of N loops:all same areaB magn. flux through one looploops in “series” emfs add!

Lenz's Law• Lenz's Law:

The induced current will appear in such a direction that it opposes the change in flux that produced it.

• Conservation of energy considerations:Claim: Direction of induced current must be so as to oppose the change; otherwise conservation of energy would be violated. » Why???

• If current reinforced the change, then the change would get bigger and that would in turn induce a larger current which would increase the change, etc..

vB

S Nv

BN S

Lecture 12, ACT 3• A conducting rectangular loop moves with constant

velocity v in the +x direction through a region of constant magnetic field B in the -z direction as shown.– What is the direction of the induced current in the

loop?

(c) no induced current(a) ccw (b) cw

x

y

Lecture 12, ACT 4•A conducting rectangular loop moves with constant velocity v in the -y direction away from a wire with a constant current I as shown.

• What is the direction of the induced current in the loop?

(a) ccw (b) cw (c) no induced current

x

y