Basic Definition

At a basic level, a magnetic field is an invisible field around a magnetic object or a moving electric charge that exerts a force on other magnetic objects or moving electric charges within the field.

Similar to electric fields, magnetic objects have something similar to a "charge" called polarity. Like poles repel, while opposite poles attract.

Visualization

Here's what the magnetic field looks like on a magnet with a north pole (-) and south pole (+):

Sources Of Magnetic Fields

There are two sources of magnetic fields:

1. Permanent Magnets

2. Electric Currents

Permanent Magnets

Imagine a bar magnet. It has a north pole and a south pole. The field lines leave the north pole and loop around to the south pole, forming a closed loop.

Electric Currents

When an electric current flows through a wire, it generates a magnetic field around it. This is described by the Right-Hand Rule: if you point the thumb of your right hand in the direction of the current, your fingers curl around the wire in the direction of the magnetic field lines.

Law of magnetic poles

Here are the laws of magnetic poles:

• Every magnet has two poles (north and south)

• Like poles repel

• Opposite poles attract

Magnetic Field in a Wire

The magnitude of the magnetic field around a wire can be determined from the following formula:

$| \vec {B} | = \frac { \mu_0 I } { 2 \pi r } \quad [T]$

Where $\mu_0 = 4 \pi \times 10^{-7} \frac {T \ \cdot \ m} {A}$ is the permeability of free space, $I$ is the current in flowing through the wire, and $r$ is the distance from the wire. The magnetic strength is in the units of Tesla ($T$), and typically $1 T$ is a lot, so you will most likely end up with smaller values.

Here's a 3D visualization:

The Right Hand Rule

You can use a variant of the right hand rule to determine the rotation of a magnetic field around a wire.

By pointing your thumb in the same direction as the current, and wrapping your fingers around, like a thumbs-up, your remaining fingers show you the dirction of rotation of the magnetic field:

Example

What is the magnetic field $1.25m$ from a wire with a current of $12.5A$ passing through it?

Answer

$| \vec {B} | = \frac { \mu_0 I } { 2 \pi r } = \frac { (4 \pi \times 10^{-7} \frac {T \ \cdot \ m} {A})(12.5A) } { 2 \pi (1.25m) } = 2 \times 10^{-6} T$

Selenoids

A selenoid is essentially a wire coiled up that produces a magnetic field similar to that of a dipole magnet going through its center.

Again, you can use the right hand rule to determine the direciton of the magnetic field lines by wrapping your fingers in the direction of the current, in which case your thumb points to the direction at which the magnetic field lines exit the selenoid.

Here's what it looks like:

As you can see, the current moves through the spiral of the selenoid. In this example, by using the right hand rule, the magnetic field would go out through the left side of the selenoid.

Magnetic Forces

Magnetic forces arise from the interaction between magnetic fields and moving charges or other magnets. These forces can act on individual charged particles, current-carrying wires, or entire magnetic objects.

When a charged particle moves through a magnetic field, it experiences a force called the Lorentz force. The direction and magnitude of this force depend on the charge of the particle, its velocity, and the strength and direction of the magnetic field.

Here is how you calculate it:

$\vec {F}_m = q(\vec {v} \times \vec {B}) \quad \text {or} \quad | \vec {F}_m | = q | \vec {v} | | \vec {B} | \sin \theta$

Where $\vec {F}_m$ is the magnetic force, $q$ is the charge of the particle, $\vec {v}$ is the velocity of the particle, $\vec {B}$ is the magnetic field and $\theta$ is the angle at which the particle passes through the magnetic field.

The magnetic force is essentially the cross-product of the velocity of the particle and the magnetic field, multiplied by the charge.

The Right Hand Rule

You can use the right hand rule to remember the directions of the vectors. In the picture below, you can use your index finger to represent the direction of the velocity of the charge, your thumb for the direction of the magnetic force, and your middle finger for the direction of the magnetic field.

Where $\vec {F}_m$ is the magnetic force, $q$ is the electric charge, $v$ is the velocity of the charge, and $B$ is the magnetic field.

Example

Calculate the acceleration of an electron that travels at a velocity of $3.15 \times 10^4 \frac {m} {s}$ east through a magnetic field of $0.295T$ pointing south. Remember, the mass of an electron is $9.11 \times 10^{-31} kg$ and its charge is $e = 1.60 \times 10^{-19} C$.

Answer

First let's calculate the force:

$\vec {F}_m = q(\vec {v} \times \vec {B}) = (1.60 \times 10^{-19} C)(3.15 \times 10^4 \frac {m} {s})(0.295T) = 1.49 \times 10^{-15} N$

Now, we can determine the acceleration

$\vec {F}_m = m_e \vec {a} \ \Rightarrow \ \vec {a} = \frac { \vec {F}_m } { m_e } = \frac { 1.49 \times 10^{-15} N } { 9.11 \times 10^{-31} kg } = 1.63 \times 10^{15} \frac {m} {s^2}$

By using the right hand rule, we can determine that the acceleration is pointed upwards.

$\therefore$ The electron is accelerating $1.63 \times 10^{15} \frac {m} {s^2}$ upwards.

Electromagnetic induction

Electromagnetic induction is the process by which a changing magnetic field within a closed loop of wire induces an electromotive force (EMF) and, consequently, an electric current in the wire.

This is what it looks like:

In the video, the red portion of the magnet is the north dipole, and the blue portion is the south dipole. You can see that as the red side moves into, then out of the selenoid, the currents follow the right hand rule. Inversely, this same principle applies when the blue side enters, then exits as well.