Magnetic Fields
Every magnetic material, regardless of size and shape, has two
“poles”: North and South.
The nature of these poles are defined by one simple set of observations.
There are forces between any two poles:
North poles repel other north poles.
South poles repel other south poles.
A north and a south pole attract.
This looks and feels rather similar to the negative and the positive
electric charge.
A magnetic pole behaves like a “magnetic charge”
EXCEPT THAT
no-one has ever been able to isolate a single magnetic pole.
They always come in pairs: North and South.
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Magnetic Fields: A Timeline
1269 - Petrus Peregrinus of Picardy, Italy, discovers that natural spherical magnets
(lodestones) align needles with lines of longitude pointing between two pole positions on the
stone.
1600 - William Gilbert, court physician to Queen Elizabeth, discovers that the earth is a giant
magnet just like one of the stones of Peregrinus, explaining how compasses work.
1820 - Hans Christian Oersted discovers that electric current in a wire causes a compass
needle to orient itself perpendicular to the wire.
1820 - Andre Marie Ampere, one week after hearing of Oersted's discovery, shows that
parallel currents attract each other and that opposite currents repel.
1820 - Jean-Baptiste Biot and Felix Savart show that the magnetic force exerted on a
magnetic pole by a wire falls off like 1/r and is oriented perpendicular to the wire.
1825 - Ampere publishes his collected results on magnetism.
Taken from Ross Spencer: “A Ridiculously Brief History of Electricity and Magnetism”
(who took it from E. T. Whittaker’s “A History of the Theories of Aether and Electricity”)
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Magnetic Fields
Examples of common magnetic materials:
Iron, cobalt, nickel
As we did with electric charges, so we do with magnetic poles.
We break up our conception of the force of interaction between magnetic
poles into the following parts:
The magnetic pole gives rise to a magnetic field around it.
That field can interact with other magnets.
A magnet cannot interact with itself.
This looks and feels rather similar to the negative and the positive
electric charge.
A magnetic pole behaves like a “magnetic charge”
EXCEPT THAT
no-one has ever been able to isolate a single magnetic pole.
They always come in pairs: North and South.
3
The Interaction Between
Electricity and Magnetism
1820 - Hans Christian Oersted discovers that electric current in a wire causes a
compass needle to orient itself perpendicular to the wire.
1820 - Andre Marie Ampere, one week after hearing of Oersted's discovery, shows
that parallel currents attract each other and that opposite currents repel.
In hindsight we will begin from an even simpler starting point.
Observation 1: There are cosmic rays that are constantly entering the earth's
upper atmosphere. These particles are positively charged. How do we know ?
Charged particles that are moving experience a force in the presence of a
magnetic field. This force has a most unusual character.
Put on your vector hats, now.
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The Magnetic Force on a Charged Particle
FB =q v x B
The force on a moving charged particle is:
Proportional to its speed
Proportional to the magnitude of the charge
Proportional to the magnitude of the magnetic field strength |B|
What is unusual is the relation between direction of the particle velocity, the
magnetic field, and the direction of the force. Here is the prescription:
Point your fingers towards the velocity vector with your thumb sticking out
(as in a handshake).
Do it in such a way that you can curl your fingers in the direction of the
magnetic field vector.
The direction of the force is the direction in which the thumb points.
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The Magnetic Force on a Charged Particle
The direction of force on a moving charged particle is given by the
Right Hand Rule. Since we are dealing with three dimensions, we
need some visualization aids.
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We denote magnetic field pointing out of the page by dots or points.
We denote magnetic field pointing into the page by crosses.
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The Magnetic Force on a Charged Particle
Lets consider a positively charged particle moving to the right.
In this case the velocity vector is perpendicular to the magnetic field.
FB=q v x B=q
v
Bsin 90 o =q v B
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The Magnetic Force on a Charged Particle
The magnitude of the force in this geometry is:
F B =FB=q v B
The direction of force on a moving charged particle is given by the
right hand rule.
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A Mechanical Analogy
An analogous mechanical situation is a ball on a string being being
swung in a horizontal circle.
Here too the direction of the velocity vector and the force vector are
perpendicular to each other.
v
FT
The result of this is circular motion.
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The Magnetic Force on a Charged Particle
When a charged particle enters a region where there is a magnetic field
that points perpendicular to the velocity vector of the charged particle,
the magnetic force is perpendicular to the velocity vector,
and the end result is circular motion
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Motion of a Charged Particle in a
Uniform Magnetic Field
Back to Newton's second law:
FB =m a
F=
where
F B =FB=q v B
In uniform circular motion, there is this funny situation:
there is acceleration even though the speed is constant, because the
direction of the velocity vector keeps changing.
the acceleration vector is perpendicular to the velocity vector
2
this acceleration has a magnitude a=v /r
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Motion of a Charged Particle in a
Uniform Magnetic Field
Back to Newton's second law:
FB =m a
F=
where
F B =FB=q v B
In uniform circular motion, there is this funny situation:
there is acceleration even though the speed is constant, because the
direction of the velocity vector keeps changing.
the acceleration vector is perpendicular to the velocity vector
2
this acceleration has a magnitude a=v /r
The acceleration points in the same direction as the force.
So we can write:
mv 2
qvB=F B =m a=
r
or
mv 2
qvB=
r
or
mv
r=
qB
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Example: A proton moving perpendicular to
a uniform magnetic field
A proton is moving in a circular orbit of radius 14.0 cm in a uniform
0.35 T magnetic field directed perpendicular to the velocity of the
proton.
Find the speed of the proton.
Find the period of circular motion of the proton.
We begin here:
mv
r=
qB
rqB
v=
m proton
r =0.14 m
q=1.6×1019 C
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We need the mass of the proton: m proton =1.67 ×10
Putting in the
numbers, we
get:
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v=0.14 1.6×10
kg
0.35 T 6
=4.7×10
m /s.
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1.67 ×10 kg
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Example: A proton moving perpendicular to
a uniform magnetic field
A proton is moving in a circular orbit of radius 14.0 cm in a uniform
0.35 T magnetic field directed perpendicular to the velocity of the
proton.
Find the speed of the proton.
Find the period of circular motion of the proton.
We begin here:
2 v
=
=
T
r
rqB
m proton
m proton
We get: T =2 qB
Since: v=
When you put in the numbers, the
answer is ~0.1 micro-seconds
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Motion of a Charged Particle in a
Uniform Magnetic Field
So far, we have only considered charged particles whose velocity
vector is perpendicular to the magnetic field vector.
What happens when the two vectors are parallel?
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Motion of a Charged Particle in a
Uniform Magnetic Field
So far, we have only considered charged particles whose velocity
vector is perpendicular to the magnetic field vector.
What happens when the two vectors are parallel?
The charged particle continues on undeflected.
The velocity is unchanged in magnitude and in direction.
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Motion of a Charged Particle in a
Uniform Magnetic Field
So far, we have only considered charged particles whose velocity
vector is perpendicular to the magnetic field vector
or parallel to the magnetic field vector.
What happens when the angle is neither 0 or 90 degrees ?
Split the vector up into two components:
One perpendicular to the magnetic field
- this component of the velocity changes in direction in a plane
perpendicular to the field direction
The other parallel to the magnetic field
- this component is unchanged.
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Motion of a Charged Particle in a
Uniform Magnetic Field
So far, we have only considered charged particles whose velocity
vector is perpendicular to the magnetic field vector
or parallel to the magnetic field vector.
What happens when the angle is neither 0 or 90 degrees ?
Split the vector up into two components:
One perpendicular to the magnetic field
- this component of the velocity changes in direction in a plane
perpendicular to the field direction
The other parallel to the magnetic field
- this component is unchanged.
What kind of motion does this particle undergo ?
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