PES 1120 Spring 2014, Spendier
Lecture 30/Page 1
Today:
- Force between two parallel, infinite wire currents
- Ampere's Law
- Solenoids
DEMO: We have seen that moving charges or currents are the source of magnetism. This can
be readily demonstrated by placing compass needles near a wire. As shown in Figure, all
compass needles point in the same direction in the absence of current. However, when I  0 ,
the needles will be deflected along the tangential direction of the circular path
Last time:
Magnetic Field of a Moving Charged Particle
 0 q v  r 0 q v  rˆ
B

4 r 3
4 r 2
Biot-Savart Law:


0 I ds  rˆ
B
4
r2
(similar to Coulomb's Law)
PES 1120 Spring 2014, Spendier
Lecture 30/Page 2
Example: A circular loop has radius R and carries current I2 in a clockwise direction. The
center of the loop is a distance D above a long, straight wire.
(a) What is the direction of the current I1 in the wire if the magnetic field at the center of
the loop is zero?
(b) Derive an equation for the magnitude of the current I1.
PES 1120 Spring 2014, Spendier
Lecture 30/Page 3
Force between two parallel, infinite wire currents:
Two long straight wires, both in the plane of the page are carrying equal currents as
shown. What is the direction of the forces that the wires exert on each other?
What is the force on wire b due to the current in wire a? The current in wire a produces a

magnetic field Ba . To find the force on wire b we need the magnitude and direction of

the field Ba at the side of wire b.
I
Ba  0 a
(d = distance between the two wires)
2 d
The curled–straight right-hand rule tells us that the direction of at wire b is down. Now
that we have the field, we can find the force it produces on wire b.

The force on a length L of wire b due to Ba is

 
Fab  I b L  Ba
for two parallel wires the magnitude of the force on each wire is

 LI I
Fab  I b LBa sin 90  0 b a
2 d
Parallel currents attract each other, and antiparallel currents repel each other.
antiparallel currents: rail guns
A railgun is an electrically powered electromagnetic projectile launcher based on similar
principles to the homopolar motor. A railgun comprises a pair of parallel conducting
rails, along which a sliding armature is accelerated by the electromagnetic effects of a
current that flows down one rail, into the armature and then back along the other rail.
Navy’s New Railgun Can Hurl a Shell Over 5,000 MPH
http://www.wired.com/2014/04/electromagnetic-railgun-launcher/
PES 1120 Spring 2014, Spendier
Lecture 30/Page 4
Analogous to Gauss’s law in electrostatics
  qencl
E  
E
  dA  0
(first term: electric flux through a closed surface)
we now introduce Ampere’s law in magnetism. In order to apply both of them, the
system must possess certain symmetry.
Properties of the Magnetic Field
The smallest source of a magnetic field is a magnetic dipole – there is still no beginning
or end to the magnetic field lines! As we know, the source of magnetic fields is a moving
charge, and in a magnet, it is the electrons in the atoms that cause the field.
So, if there is no source or sink of magnetic fields, there can be no NET flux through a
closed surface (every field line that enters a closed surface must eventually exit the closed
surface)!
Then, given what we understand about Gauss’s Law for the electric field, we can deduce
that:
 
 B  dA  0 (Gauss’s Law for Magnetic Fields)
In other words, there is no magnetic charge (magnetic monopoles).
Another important property of the magnetic field can be seen by noticing that the
magnetic field lines always form loops around currents:
We can write for an infinite current-carrying wire:

Closed Path
 
B  dl 

B dl cos(0) 
Closed Path

B dl
Closed Path
Magnetic field produced by infinite wire a distance r away: B 
differential arc length: dl  rd
gives
I 0
2 r
PES 1120 Spring 2014, Spendier
2

B dl 
0 I
 2 r
rd 
0
Closed Path
Lecture 30/Page 5
0 I
2  0 I
2
What if the path doesn’t enclose the current?

Closed Path

Closed Path

 
B  dl 

 
B  dl 

B dl cos(0) 

B dl  0 
 
B  dl 
ab
 
B  dl 
ab
Closed Path
ab

 
B  dl 
bc

 
B  dl 
c d

 
B  dl
d a
B dl cos(90) 
b c



B dl cos(180) 
c d

B dl cos(90)
d a
B dl  0
c d
 /2
   / 2 0 I
0 I
B

dl

r
d



0 2 r2 2
0 2 r1 r1 d  0
Closed Path

So, we found that for two different closed paths, the integral of B dotted into the path

element dl gave two different answers. We can encompass both ideas into one equation
which is known as Ampere's Law
Ampere’s Law
 
B
  ds  0 I encl
The little circle on integral indicates that the integrate is over a line which closes on itself.
Ampere's law is fundamental, but incomplete at this point. We will have to return to that
later.
Ampere’s law is applicable to the following current configurations:
1. Infinitely long straight wires carrying a steady current I
2. Infinitely large sheet of thickness b with a current density J
3. Infinite solenoid
4. Toroid
We will examine some of these configurations in detail.
PES 1120 Spring 2014, Spendier
Lecture 30/Page 6
In the case of an infinite wire, the system possesses cylindrical symmetry and Ampere’s
law can be readily applied as shown above. However, when the length of the wire is
finite, Biot-Savart law must be used instead.
Applying Ampere’s Law
We can use Ampere’s Law now to calculate the magnetic field from certain current
configurations.
Field Inside and Outside a Current-Carrying Wire
Consider a long straight wire of radius R carrying a current I of uniform current density,
as shown in the Figure. Find the magnetic field everywhere.
a) Outside the wire where r  R , the Amperian loop completely encircles the current.
Hence applying Ampere's law:
 
B
  ds  0 I encl  0 I
2
 B dl
cos(0)  B  rd  Br 2
0
B  2 r   0 I
0 I
2 r
direction from RHR: counter-clock wise
Same result as we obtained using the Biot-Savart Law.
B
b) Inside the wire where r  R , the amount of current encircled by the Amperian loop is
proportional to the are enclosed, i.e.,
  r2 
I enc  
I
2 
 R 
PES 1120 Spring 2014, Spendier
Lecture 30/Page 7
 
 r2 
B

ds

B
2

r


I


  0 encl 0  2  I

R 
2
r I
 Ir
B 0 2  0 2
2 rR
2 R
We see that the magnetic field is zero at the center of the wire and increases linearly with
r until r = R. Outside the wire, the field falls off as 1/r. The qualitative behavior of the
field is depicted in Figure below:
Solenoid
A solenoid is a long coil of wire tightly wound in the helical form. The figure below shows
the magnetic field lines of a solenoid carrying a steady current I. We see that if the turns are
closely spaced, the resulting magnetic field inside the solenoid becomes fairly uniform,
provided that the length of the solenoid is much greater than its diameter. For an “ideal”
solenoid, which is infinitely long with turns tightly packed, the magnetic field inside the
solenoid is uniform and parallel to the axis, and vanishes outside the solenoid.
The right figure indicates that the magnetic field of a solenoid looks like the magnetic field of
a permanent Bar magnet. However, a permanent magnet can never be turned off. The
magnetic field of a solenoid can be turned off simply by turning the current off. Solenoids
form the basis of many "electromagnets" that can be controlled by currents.
PES 1120 Spring 2014, Spendier
Lecture 30/Page 8
Application for a solenoid: Speakers that convert an electric signal (current) into sound we
hear.
A speaker's function is fairly simple. A speaker converts electrical signals into acoustical
energy: sound. By moving back and forth, the speaker increases and decreases the air
pressure in front of it thus creating sound waves.
The voice coil and standing magnetic field are what makes the cone move. When an
electrical current/signal from an audio amplifier (alternating at the same frequency as the
sound that created it) is put into a speaker's voice coil, the voice coil generates an
alternating magnetic field. The alternating polarity of the voice coil causes it to be
repelled-from and attracted-to the standing magnetic field. This action of alternating
attracting and repelling causes the cone (attached to the voice coil) to move! When the
cone moves between 20Hz-20 KHz, the speaker makes sound that we can hear!