# Magnetism

Cloud chambers are a simple method of detecting high energy charged particles. In order to detect the particles, the chamber is filled with a supersaturated vapor of water or alcohol. High speed particles disrupt the vapor and leave a trail.

As radiation passes through the chamber only charged particles are visible. This is because neutral particles don't disrupt the vapor. When a powerful magnetic field is applied to the cloud chamber the charged particles curve. The arc of their curve indicates charge and mass.

# Magnetic Fields

Magnetic fields are a modification to electric fields that comes from special relativity. Like electric fields, magnetic fields are produced by charged particles, but only when the charged particle is moving.

Moving charges produce magnetic fields.

A moving charge produces a magnetic field, but a moving charge also experiences a force from external magnetic fields.

Magnetic fields apply a force on moving charges.

For example, a moving electron produces a magnetic field. This field applies a force on other moving electrons.

Strong magnetic forces are produced through electromagnetism and ferromagnetism. Weaker forms of magnetism also exist, like diamagnetism and paramagnetism.

## Magnetic Force on a Particle

The magnetic field in the diagram is directed out of the page. This field produces a force on moving charges. The direction of the magnetic force is inverted for negative charges.

## $$F=qvB \sin \theta$$

$$F$$ = magnetic force [N, newton, kg m/s²] vector
$$q$$ = charge [C, Coulomb]
$$v$$ = velocity [m/s] vector
$$B$$ = external magnetic field [T, tesla, kg/C/s] vector
$$\theta$$ = angle between v and B

What does ⨂ and ⊙ mean?
= into the page
= out of the page

To show 3D vectors we will use these symbols for into and out of the page. I think of the cross as the feathers on the back of an arrow, and the dot as the tip of an arrow.

Magnetic fields are measured in teslas, which can have a wide range of values.

Example: A +2 C particle is moving at 10 m/s through a 5 mT field from a refrigerator magnet. The velocity and field are at a right angle to each other. What magnitude force will be applied to the particle?
solution $$F=qvB$$ $$F = (2)(10)(0.005)$$ $$F = 0.1 \, \mathrm{N}$$
Example: To measure the strength of a magnetic field, electrons are fired into the field at 100 m/s. The magnetic field accelerates the electrons at 255 m/s². What is the magnitude of the magnetic field perpendicular to the particle's velocity?
solution $$F = ma$$ $$F = (9.1 \times 10^{-31})(255)$$ $$F = 2.32 \times 10^{-28} \, \mathrm{N}$$
$$F = q v B$$ $$B = \frac{F}{qv}$$ $$B = \frac{2.32 \times 10^{-28}}{(1.6 \times 10^{-19})(100)}$$ $$B = 1.45 \times 10^{-11} \, \mathrm{T}$$

## Direction of Magnetic Forces

The 3D nature of the magnetic force can be solved with a vector operation called a cross product. The direction of a cross product can be found with the right hand rule.

Using your right hand, place your thumb in the direction of the current/charge. Place your index finger in the direction of the magnetic field. The palm of your hand will face in the direction of the force.

Example: What direction is the magnetic force on the charged particle?
solution

Using the right hand rule place your thumb towards the top of the page and your fingers to the right.

$$\text{into the page}$$
Example: A magnetic field is pointed north. A positively charged particle in the field is moving down, towards the ground. What direction is the magnetic force on the particle?
solution

Using the right hand rule place your thumb down and your index finger north.

$$\text{east}$$
Example: What direction magnetic field would produce a downward force on a negatively charged particle moving west?
solution

A negatively charged particle will go in the opposite direction of a positively charged particle. Using the right hand rule place your thumb west east and the palm of your hand down.

$$\text{south}$$
Example: A 1.0 C charged particle is moving west at 10 m/s through a 2 T magnetic field directed east. What is the magnitude and direction of the force on the particle?
solution

Since the velocity and magnetic field are in opposite directions there is no force.

$$F=qvB \sin \theta$$ $$\sin 180 = 0$$ $$F=0$$
Example: Find the magnitude and direction of the force on a 24 μC charge moving down at 13 m/s through an 8.5 T magnetic field directed south.
solution $$F = q v B$$ $$F = (24 \times 10^{-6})(13)(8.5)$$ $$F = 0.00265 \, \text{N west}$$

## Magnetic Force on a Wire

Running electric current through a wire moves a huge number of charges. Each charge follows F=qvB, but we can treat them as a whole with the concept of electric current.

derivation of magnetic force on a wire

We can substitute the definition of electric current and velocity.

$$I = \frac{q}{\Delta t} \quad \quad v = \color{teal}\frac{\Delta x}{\Delta t}$$ $$q = \color{magenta}I \Delta t \quad \quad \quad \quad \quad \, \,$$
$$F=qvB$$ $$F= {\color{magenta}(I \Delta t)} {\color{teal}\left( \frac{\Delta x}{\Delta t} \right )} B$$ $$F= I \Delta x B$$

We will use l for length instead of Δx.

$$F= I l B$$

A wire in a magnetic field will experience a force when current runs through it.

## $$F=IlB \sin \theta$$

$$F$$ = magnetic force [N, newton, kg m/s²] vector
$$I$$ = current [A, Amps]
$$l$$ = length of wire in the magnetic field [m, meter] vector
$$B$$ = magnetic field [T, tesla, kg/C/s] vector
$$\theta$$ = angle between the magnetic field and the current

Note that current is in the opposite direction of the flow of electrons, since electrons are negative.

The magnetic force can be used to build a rail gun.

Example: A 15 cm wire has 20 mA flowing through it east to west. The wire is in a magnetic field of 10 T directed up. Describe the force on the wire.
right hand rule
solution $$F=IlB$$ $$F = (0.02)(0.15)(10)$$ $$F = 0.03 \, \text{N north}$$
Example: A magnetic field is pointed to the right. Electric current is flowing towards the top of the page. What direction is the magnetic force on the wire?
solution

Using the right hand rule place your thumb towards the top of the page and your fingers to the right.

$$\text{into the page}$$
Example: How long is a wire that has a force of 0.050 N caused by the 0.21 A current interacting with a 0.45 T magnetic field at a right angle?
solution $$F=lIB$$ $$l= \frac{F}{IB}$$ $$l= \frac{0.050}{(0.21)(0.45)}$$ $$l= 0.53 \, \mathrm{m}$$

# Electromagnets

Electric current measures the flow of charge. The current in a straight wire produces a magnetic field that curves around the wire.

What does ⨂ and ⊙ mean?
= into the page
= out of the page

To show 3D vectors we will use these symbols for into and out of the page. I think of the cross as the feathers on the back of an arrow, and the dot as the tip of an arrow.

The electric current from a long wire forms a magnetic field in the shape of concentric circles around the wire. The field strength diminishes with distance at 1/r not 1/r². The field strength can be calculated with Ampere's law.

The right hand rule can be used to find the direction the field turns around a wire. Place your right hand's thumb in the direction of the current. Your finger will curl in the direction the field turns.
Example: Curl the fingers on your right hand with the magnetic field to determine the direction of the current?
solution

Your thumb should point down. The current is directed towards the bottom of the page.

Two wires are aligned parallel at a constant distance of 0.3 m from each other. Each wire has 2.0 A of electric current moving towards the top of the page.

Example: What direction is the magnetic field produced by the current in the left wire at the location of the right wire?
solution

The field produced by the left wire is into the page on the right wire.

Example: What about the direction of the field from the right wire on the left?
solution

The field produced by the right wire is out of the page on the left wire.

Example: Use each wire's magnetic field to determine the direction of the forces between the wires.
solution

Wires with current in the same direction are pulled towards each other.

A loop of current produces a magnetic field with a north and south pole, like a ferromagnet.

The magnetic field from one wire can be increased by wrapping the wire in many loops. The strength can increased again by wrapping the wire around a ferromagnetic material, like a nail.

Electromagnets are used in most devices that produce motion from electricity: electric motors, speakers, microphones, valves, ...

They are also used in generators, MRI machines, transformers, induction heating, magnetic levitation ...

Electromagnetic induction is the production of voltage from a conductor moving past a magnetic field.

The process is used to make voltage from motion with a generator.
Example: Nuclear, coal, hydroelectric, natural gas, and wind power

# Ferromagnetism

Ferromagnetic metals produce a lasting magnetic field after they are exposed to an external magnetic field. Ferromagnetic metals include: iron, nickel, cobalt, most of their alloys, and rare earth metals.

Experiment: Test what an magnet is attracted to.
observations

Paper clips, staples, forks, screwdrivers, hammers, nails, scissors, refrigerators, cars, pans, speakers, microphones, batteries, phones.

Experiment: Put two magnets near each other. What do you observe?
observations

One pair of ends attract and one pair repels.

We say that a magnet has a north and south pole. North and south poles attract, while two of the same poles repel.

If you break a magnet, each smaller piece also has a north and south pole. It's impossible to have a single north or south pole, a monopole.

You can see a 2D slice of the 3D field from a magnet by sprinkling small bits of iron around the magnet. The field has north and south poles just like electromagnet.

Electrons do something similar to spinning around, but with some quantum mechanical differences. This "spin" is the source of magnetic fields in ferromagnetic metals.

Two electrons normally pair up and spin in opposite directions. This pairing cancels out most large scale magnetic effects. But in ferromagnetic materials some spins remain unbalanced. The unbalanced spinning electrons can produce a net magnetic field, but only if the material has its domains aligned as well.

Question: What direction are the field lines at the north pole of a ferromagnet or an electromagnet?

The north pole has field lines that are directed away from the magnet. The south pole as the same number of field lines directed into the magnet.

Question: What direction is the force the magnet is exerting on the wire?

Using the right hand rule:

the magnetic field is out of the north pole, to the right
the electric current is up
the force is into the page

Question: What direction is the force the magnet is exerting on the wire?

Using the right hand rule:

the magnetic field is out of the north pole, to the right
the electric current is down
the force is out of the page

Question: What direction is the force the magnet is exerting on the wire?

This one is tricky. The field in the center of the wire is parallel with the current, so there is no force.

The field at the left side of the magnet has a small component directed up. This produces a force on the left side of the wire directed out of the page. The field on the right side of the wire is into the page.

These uneven forces on the sides of the wire will make it spin.

Question: What direction is the force the magnet is exerting on the wire?

Using the right hand rule:

the magnetic field is into the south pole, to the right
the electric current is up
the force is into the page

## Earth's Magnetic Field

The Earth produces a magnetic field that is closely aligned with its axis of rotation. The source of the field is from convection of conductive materials in Earth's core, but the process is complex and has only been recently modeled with computer simulations.

The Earth's magnetic field slows down charged particles. This shields the Earth from the solar wind's ionizing radiation. This slowing down can be seen at the north and south poles as an aurora.

We have strong evidence that the north and south poles experience periodic reversals. About 780 000 years ago a compass would have pointed in the opposite direction it points now. The reversals are unpredictable, but typically there is thousands of years of stability between flips.

A compass has a small magnet that aligns with the Earth's magnetic field.

Question: The north pole of a magnet points towards the geographic north pole in the arctic circle. Does this make sense?

This doesn't make sense. Two north poles would repel.

The geographic north pole is actually the Earth's magnetic south pole.
So confusing!

## Motion in a Magnetic Field

The magnetic force on a charged particle in a uniform magnetic field is always at a 90 degree angle to the velocity. This satisfies the requirement for circular motion and it means we can set the equation for magnetic force equal to the centripetal force.

derivation: the motion of a charged particle in a magnetic field $$F = qvB \quad \quad F = \frac{mv^{2}}{r}$$ $$qvB = \frac{mv^{2}}{r}$$ $$qB = \frac{mv}{r}$$ $$r = \frac{mv}{qB}$$

## $$r = \frac{mv}{qB}$$

The Earth's magnetic field protects us from high speed charged particles. Most of these particles come from solar wind.

Example:What is the radius for the circular motion of an electron in the Earth's magnetic field? Wikipedia says that the field is up to 70 μT, and particles from the solar wind travel at up to 500 km/s.
solution $$r = \frac{mv}{qB}$$ $$r = \frac{(9.1 \times 10^{-31})(500\,000)}{(1.6 \times 10^{-19})(70 \times 10^{-6})}$$ $$r = 0.0406 \, \mathrm{m}$$
Example: Find the radius for a proton in the same conditions.
solution $$r = \frac{mv}{qB}$$ $$r = \frac{(1.67 \times 10^{-27})(500\,000)}{(1.6 \times 10^{-19})(70 \times 10^{-6})}$$ $$r = 74.55 \, \mathrm{m}$$