Magnetic Forces and Torques

Review:

Charged Particle in an external field:

Straight wire of length L with current I in a uniform external field B:

BvqF

F I L B

Note: L points in direction of positive current flow

If B not uniform, and/or wire not straight: the force dF on a short segment of vector length dL is

I

I

Segment of length dL

The total force on the wire is:

alongwire

F IdL B ������������������������������������������

dFB

dL

dF = IdL x B

Force on a current-carrying wire (general case)

Example 1

y

x

x x x

x x x

x x x

I

(uniform)

R

Find the force on:

a) The straight wireb) The semicircular

wirec) The whole circuit

For (b): start with force dF due to an infinitesimal piece, and do the integral.

B

Solution

Total magnetic force on the loop = 0

Proof:

F IdL B

I dL B

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But: for a closed loop, so F = 0

0dL����������������������������

Theorem: For any closed current loop in a uniform magnetic field,

(if B is a constant vector)

Torque:

Although there is no net force on a circuit in a uniform field, there may be a net torque. The torques due to equal and opposite forces applied at different locations do not necessarily cancel.

We can calculate the torque directly for a rectangular loop. There is also a simple rule, which actually applies to a plane loop of any shape. First we need to define the “magnetic dipole moment” (a vector) for a current loop.

Magnetic Dipole Moment

(or “magnetic moment”)

Define the magnetic moment of a small current loop by

AI

loop) of area" vector" ( A

Area of loop

surface) (

Note the right-hand rule! – fingers follow I, thumb points in μ

current

Torque on a Current Loop (Uniform B)

w

h

B

A B

DC

Forces:

x

FAB= I w B sinθ

FCD= I w B sinθ

FBC= I h BFAD= I h B

Example: a rectangular loop

Top view:

I

FBC= I h B

A

B

FAD = I h B

w•sinθ

w

Torque (about any pivot; e.g., at A)

sin( ) sin

( ) sin

IhB wIhw B

I area B B

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I into the page

x

��������������

B��������������

B��������������

The torque on a current loop with magnetic dipole moment in a uniform external field is: B

B

Where μ = IA

Quiz

Which direction does the torque vector point in for a flat horizontal coil on the surface of earth ?

A) NorthB) EastC) WestD) UpE) Down

50oB��������������

B��������������

I

north

Example 2

north

Circular loop, 2 turns, R = 1 m, I = 20 A, CCW (from above)

Find: Torque (magnitude and direction)

Find the torque on a flat, horizontal, circular coil due to the magnetic field of the Earth.

50o

0.5 x 10-4 T

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B��������������

Solution

Summary

Charged Particle:

Wire:

Straight wire, Uniform B:

Torque:

BvqF

or dF IdL B F IdL B ����������������������������

BLIF

B