PES 1120 Spring 2014, Spendier
Lecture 27/Page 1
Today:
- Magnetic Force on a current-carrying Wire
- Torque on a current loop
- Magnetic dipole moment
- Next Week Monday Review, Wednesday – EXAM 2
Last time we studied the consequences and applications of a moving charged particle in
magnetic (and electric) fields.
Magnetic Force on a Current-Carrying Wire
We have seen that a charged particle moving through a magnetic field experiences a

magnetic force FB . Since electric current consists of a collection of charged particles in
motion, when placed in a magnetic field, a current-carrying wire will also experience a
magnetic force.
Consider a long straight wire suspended in the region between the two magnetic poles. The
magnetic field points out the page and is represented with dots (•). It can be readily
demonstrated that when a downward current passes through, the wire is deflected to the left.
However, when the current is upward, the deflection is rightward, as shown in Figure
To calculate the force exerted on the wire, consider a segment of wire of length l
F  qvB sin f
l 
F  q   B sin f
 t 
∆q is the net charge through length l
 q 
F    lB sin f
 t 
F  IlB sin f
 

F  I l B
PES 1120 Spring 2014, Spendier
Lecture 27/Page 2

RHR for Force on Wire: Start fingers in direction of I so that they can curl towards B ,

thumb points in direction of F .
Magnetic Force on a Current : One common use of this force is in speakers.
General Calculations
For a wire of arbitrary shape, the magnetic force can be obtained by summing over the forces
acting on the small segments that make up the wire.

A curved wire has infinitely many different tangent lines  length vectors dl associated
with each point.
The force at each point on the wire:
 

dF  I dl  B
Net force on wire is found by adding (vectorially) the force at each point together.
 

F   I dl  B
Force on a current loop:
Current flowing around a closed path is called a current loop. Shown is a square current
loop in an upwards magnetic field. What is the magnitude and direction of the force
acting on side 1, 3, 2, and 4? What is the total force acing on the loop?
PES 1120 Spring 2014, Spendier
Lecture 27/Page 3

RHR for Force on Wire: Start fingers in direction of I so that they can curl towards B ,

thumb points in direction of F .
The net force on this current loop is zero.
The net force on any shape current loop in a uniform magnetic field is zero!
Torque on a Current Loop
But what about the torque on the current loop? The net torque is non-zero. This loop
would rotate In an external, uniform magnetic field, a current loop experiences zero net
force but a non-zero torque.


Even though the net force on the loop vanishes, the forces F1 and F3 will produce a torque
which causes the loop to rotate about the x-axis.
  
From Physics 1: t  r  F
(r – lever arm and at end of lever arm force is applied)
PES 1120 Spring 2014, Spendier
Lecture 27/Page 4
The torque with respect to the center of the loop is
   
        
t  r1  F1  r2  F2  r3  F3  r4  F4  r1  F1  r3  F3
Application – DC Motor
•We can now understand how a DC motor works.
•We put a current loop between the poles of a magnet.
•The torque on the loop causes rotation.
•We have to switch the direction of the current every time it rotates to keep it rotating in
one direction.
PES 1120 Spring 2014, Spendier
Lecture 27/Page 5
Area vector:

It is convenient to introduce the area vector A  Anˆ where n̂ is a unit vector in the direction
normal to the plane of the loop. The direction of the positive sense of n̂ is set by the
conventional right-hand rule. The direction of n̂ is given by curling the fingers in the
direction of the current and the thumb pointing in the direction of n̂ .
Magnetic Dipole Moment:
Using the area vector, there is one more physical quantity which is worth mentioning, it
is the magnetic dipole moment defined as μ.


mI A
[A m2]

Thee magnetic dipole moment is a vector, m ! Again, the right-hand rule determines the
direction of the magnetic moment of a current-carrying loop. This is also the direction of
the loop's area vector. We can now rewrite the torque on a current loop as:
   

t  I A B  m  B
t  ( IA) B sin q  m B sin q



θ = angle between m (or A ) and B
PES 1120 Spring 2014, Spendier
Lecture 27/Page 6
Magnetic Force on a Magnetic Moment
Now, the atoms of iron (and other metals) have a magnetic moment because of the
current loops of the electrons. In non-magnetized objects, they are randomly oriented.
But in a magnet, they are (mostly) aligned with each other, giving a net total magnetic
moment.
In a magnetic field the torque on a bar magnet tends to align the magnet's dipole moment
with the direction of the B field. Magnetic Dipoles in a magnetic field rotate to make θ =
0◦.
For a non-magnetized object, like a nail, the field polarizes (rotates) the moments in the
direction of the field. Then the nail has a net magnetic moment and is attracted to the
magnet.
When the field is removed, the atoms randomize their magnetic moments again, and the
object returns to be non-magnetized. UNLESS, you heat and then cool it, or shock it by
hitting it with a hammer…
PES 1120 Spring 2014, Spendier
Lecture 27/Page 7
Potential Energy:

The equation for torque exerted on an magnetic dipole moment m in the presence of an

magnetic field B
  
t  m B
is analogous to
  
t  p E ,


the torque exerted on an electric dipole moment p in the presence of an electric field E .
Recalling that the potential energy for an electric dipole is
 
U  pE ,
a similar form is expected for the magnetic case.
 
U  m  B
Example: What is the change in energy when a dipole with μ = 1 A · m2 flips from being
opposite to a B = 1 T external field to being parallel to it? Hint: μB = (1A · m2)(1 T) = 1J.