Lecture 5

A moving charge creates a magnetic field.
Magnetic materials have fascinated people for a long time.
Certain materials are magnetic:
- Iron, cobalt, nickel, and gadolinium are prominent.
- Some alloys can be manipulated to be magnetic.
Magnets can attract or repel each other without touching.
Opposites attract, likes repel.
Small compasses align themselves with a magnet's field.
Magnets were historically used as navigational tools.
- Allowed navigation when landmarks or stars were not visible.
- Magnets point in a specific direction, the north-seeking pole.
The Earth has a magnetic field.
- The geographic North Pole is the magnetic South Pole, and vice versa.
- A giant magnet in the Earth's center provides orientational guidance.
A compass or magnet points north due to the Earth's magnetic field.
The magnetic field is denoted by the letter $B$ and is a vector.
Iron filings are used to visualize magnetic fields.
- They align along the field lines around a magnet.
Magnetic fields go from north to south.
Horseshoe magnets also have fields going from north to south.
The field between a north and south pole resembles the field between positive and negative electric charges.
Electric and magnetic fields are related, but charges need to be moving to be affected by a magnetic field.
Demonstration: magnets deflect an electron beam.
- Electrons are made visible by a fluorescent card.
- The beam bends when magnets are brought near, showing the effect of a magnetic field on moving charges.

If the velocity of a charge and the magnetic field are perpendicular, the force is perpendicular to both.
This is an intrinsically three-dimensional concept.
If velocity and field are in the same direction, there is no force.
The force is calculated using the cross product:
- $F = q(v \times B)$
- $F$ is the force vector.
- $q$ is the size of the charge.
- $v$ is the velocity vector.
- $B$ is the magnetic field vector.
To find the direction of the force:
- Point fingers in the direction of the velocity.
- Curl fingers from the velocity vector into the magnetic field vector.
- The thumb points in the direction of the force.
Units:
- $\text{B}$ (magnetic field) is measured in Tesla (T).
- Force is in Newtons (N) when charge is in Coulombs (C) and velocity is in meters per second (m/s).
- Teslas are large units.
- MRI machines have fields of a few Tesla.
Conventions for representing the third dimension:
- Dots represent the field coming out of the page.
- X's represent the field going into the page.
Example: Velocity going up, field going into the page, force is to the left.
A charge coming out of the screen with a magnetic field pointed up will experience a force to the left.

Moving charges are commonly seen in currents in wires.
Force equation:
- $F = qvB$
- If $v = l/t$ , then $F = qlB/t$
- Since current $I = q/t$ , then $F = ILB$
- A length $l$ of wire carrying current $I$ in a field $B$ experiences a force.
Demonstration:
- A wire between the poles of a magnet carries a current.
- A battery, switch, and wire are used.
- When the switch is closed, the wire jumps.
- Reversing the current reverses the direction of the jump.
Magnets interact with wires, suggesting wires have magnetic fields.
Iron filings around a wire align in circles, indicating a magnetic field.

The magnetic field around a wire goes in circles.
Right-hand rule #2:
- Thumb in the direction of the current.
- Fingers curl in the direction of the magnetic field lines.
The magnetic field is proportional to the current and inversely proportional to the distance from the wire.
- $B \propto I/r$
- $B = \frac{\mu_0 I}{2 \pi r}$
- $\mu_0 = 4 \pi \times 10^{-7} \text{ Tesla meters per Ampere}$
Problem: A hairpin-shaped wire with current.
- The segments repel each other.
- The top wire creates a field into the page at the bottom wire.
- $v \times B$ results in a downward force.
Demonstration: Parallel wires connected to a battery.
- In series (hairpin), they spread apart.
- In parallel, they pull together.
If the current doubles, the force goes up four times.
- Double the current doubles the field, and double the current again doubles the force.
- $2 \times 2 = 4$
Looping a wire creates a larger magnetic field in the center.

Looping a wire multiple times creates a solenoid.
A solenoid is like a spring with current flowing through it.
Current flowing through the wire generates a magnetic field.
The field lines loop around, creating a magnet.
The magnet is on demand; turn the switch on, there's a magnet; turn it off, no magnet.
Right-hand rule #3: Put your fingers in the direction of the current; your thumb will be in the direction of the North Pole.
Battery-operated doorbell:
- Closing the switch activates the battery.
- Current flows through the electromagnet.
- The magnetic field pulls on a piece of metal.
- Inertia causes the hammer to strike the bell even after the circuit is broken.
- The hammer springs back and makes contact again, repeating the process until the switch is released.
Ampere's Law:
- Analog to Gauss's Law.
- Gauss's Law discovers charge; Ampere's Law discovers current.
- Ampere's Law uses $\mu<em>0$ , analogous to $\epsilon</em>0$ in Gauss's Law.
- Ampere's Law adds up field parallel to a line segment.
- Uses a two-dimensional closed loop.
- Important law, important piece of physics.

A moving charge creates a magnetic field.
Electrons have spin and angular momentum around an axis.
Spinning charge creates a magnetic field.
Electrons pair up with opposite spins, canceling out magnetic fields.
Certain elements have unpaired spins in internal electron shells.
These unpaired spins give rise to natural magnetism.
Iron, cobalt, and other atoms act as tiny atomic magnets due to unpaired spins.
If atoms are not mutually aligned, you have a ferromagnetic material but not a permanent magnet.
To create a permanent magnet:
- Melt the iron.
- Apply an external magnetic field while molten.
- Cool the iron into an ingot.
- The magnetic field aligns the atoms.
Magnetic materials become magnetic by aligning magnetic domains.
Unaligned regions result in unmagnetized materials.
Aligned regions result in a magnet.
Domains can be visualized by electron microscopy.

As magnetic domains align, they make noise.
A supersensitive microphone can pick up the vibrations.
The noise is only present when an external magnet is used to align domains.
This is because the moving charge creates a magnetic field.
The lecture covered magnetic facts, forces on moving charges and wires, and fields produced by wires.
Three new equations were introduced.