Magnetism

The magnetic field (B-field) is a property of space that is created by the motion of electric charges. We focus separately on (1) The forces magnetic fields exert on moving charges and (2) The magnetic fields created by moving charges

3D Representations

When considering magnetic fields and magnetic forces, we must use three-dimensional representations.

Notation for vectors pointing in or out of the screen:

$$ \begin{aligned}&{\text{Into the screen:}}&& \bigotimes \\&{\text{Out of the screen:}}&& \bigodot \end{aligned} $$

Magnetic Force

When a test charge or "charge of interest" is moving through a magnetic field, a magnetic force is exerted on the test charge.

The direction of the magnetic force is determined using the Magnetic Force right hand rule.

Magnitude of magnetic force acting on a moving charge

$$ |\vec{F}_B|=|q_t|v_\perp B = |q_t| v B_\perp=|q_t| v B \sin{\theta} $$

The force of a uniform B field on a moving charge will cause it to travel in a circular path of radius r:

$$ r=\frac{mv_\perp}{|q|B} $$

Magnitude of magnetic force acting on a current-carrying wire of length L

$$ |\vec{F}_B|=I_\perp L B = I L B_\perp=ILB \sin{\theta} $$

Creating Magnetic Fields

Moving charges create magnetic fields (which affect moving test charges).

Magnetic field from an infinite current-carrying wire:

Magnitude of magnetic field at distance r from the wire

$$ |\vec{B}|=\frac{\mu_0 I}{2 \pi r} $$

Direction: Grip right hand rule

Right hand grip rule: Either (1) point thumb along current, fingers curl in direction of B-field, or (2) curl fingers in direction of current loop, thumb points in direction of B-field

Magnetic field inside a current-carrying solenoid with n=N/L turns:

Magnitude of magnetic field inside solenoid

$$ |\vec{B}|=\mu_0 n I $$

Magnetic force exerted by one current-carrying wire on another*:

Force exerted by Wire 1 on a Wire 2 of length L

$$ |\vec{F}_\text{1 on 2}|=\frac{\mu_0 I_1 I_2 L}{2 \pi r} $$

*Assuming direction rules about magnetic force on a wire are satisfied

Vacuum magnetic permeability

μ₀=4π×10^-7 T‧m/A²