Magnetic Forces

Different magnetic fields can be produced from passing current through a conductor. By changing the shape of the conductor, we can vary the strength of the magnetic field formed.

Force around a long straight wire

When a current is flowing through a straight wire, a magnetic field is formed around the wire:

We can use the right-hand grip rule to determine the direction of the magnetic field around any wire. Simply point your thumb in the direction of the conventional current and wrap your fingers around the wire. The direction in which your fingers bend is the direction of the magnetic field around the wire.

The flux density at a point within the magnetic field produced by the wire is given by: $$ \begin{equation}\begin{aligned} B=\frac{\mu_0 I}{2\pi r}\\ \end{aligned}\end{equation} $$ where $\mu_0$ is the permeability of free space ($4\pi \times 10^{-7}TmA^{-1}$), $I$ is the current, and $r$ is the distance of the measurement point away from the wire.

Force around a flat circular coil

We can create a stronger magnetic field if we wrap the wire into a flat coil:

The flux density at a point within the magnetic field produced by the coil is given by: $$ \begin{equation}\begin{aligned} B=\frac{\mu_0 NI}{2r}\\ \end{aligned}\end{equation} $$ where $N$ is the number of turns in the flat coil.

Force around a long solenoid

Whereas a flat circular coil makes the current to flow in a single plane, we can create a long solenoid shape. The magnetic field will be dependent on how compact the coil is (turns per unit length):

The flux density at a point within the magnetic field produced by the solenoid is given by: $$ \begin{equation}\begin{aligned} B=\mu_0 nI\\ \end{aligned}\end{equation} $$ where $n$ is the number of turns per unit length of the solenoid, measured in per metres ($m^{-1}$): $$n=\frac{N}{L}$$