Sources of Magnetic Fields

There are two sources of magnetic fields:

• Permanent magnet: The magnetic field is generated by the internal structure of the material itself.

• Electromagnet: The magnetic field is generated by the application of an electric current.

All magnets are dipoles, meaning that they have both a north and south pole. Like poles repel, opposite poles attract.

Magnetic Force

The magnetic force in a particle is equal to the charge of the particle times the cross product of its velocity and subjected magnetic field:

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The magnitude of the force is thus:

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Magnetic Field Lines

Magnetic field lines help to visualize the presence of a magnetic field. These lines form a closed path from the north to south pole of a magnet. The greater the density of lines, the stronger the field.

Magnetic Force on a Current Carrying Wire Subjected to an External Magnetic Field

The magnetic force of a wire carrying a current subjected to a uniform magnetic field is given by:

Note that the length vector points in the direction of the current’s conventional flow.

Ampere’s Right-Hand Rule

To find the direction of the magnetic field created by the current flowing through a wire, point your thumb in the direction of the current  and wrap your fingers around it. The wrapped direction of your fingers corresponds to the flow of the generated magnetic field. The direction of the magnetic field at a given point around the wire is its tangential path intersecting the point.

Electromagnetic Induction

Induced current and induced electromotive force (EMF) are two phenomena caused by a change in the amount of magnetic field passing through a coil. These quantities are governed by the rate at which the magnetic field changes.

• Induced Current: The current produced in the coil by the changing magnetic field.

• Induced Electromotive Force (EMF): The work done per unit charge to produce an induced current. The quantity is not actually a force, but a difference in electric potential (voltage drop).

• Induction: The process of producing induced current and induced EMF.

Magnetic Flux

The magnetic flux quantifies the amount of magnetic field passing through a surface. The magnetic flux through a wire loop with an area A subjected to a magnetic field is given by:

Note that n-hat is the normal vector perpendicular to the area of the loop of wire.

Faraday’s Law

We can quantify electromagnetic induction by calculating the induced EMF using Faraday’s law:

Here N is the number of coils/loops of wire. We assume that the coils (or loops) of wire are closely spaced so that the flux through them is the same. The negative sign indicates that the EMF wants to oppose the change in the magnetic field.

Lenz’s Law

An induced current has a direction such that the magnetic field due to the induced current opposes the change in magnetic flux that induces it.

• If an external magnetic field is increasing in a particular direction, its induced magnetic field will point in the opposite direction.

• If an external magnetic field is decreasing in a particular direction, its induced magnetic field will point in the same direction.

To find the direction of the induced current, we can use our Right-Hand Rule: Point your thumb in the direction of the induced magnetic field with your palm faced at the center of the coil. The curled path of your fingers will point in the direction of the induced current.

Motional EMF

The movement of a conductor in a magnetic field induces an electromotive force. This is quantified by the following expression:

Fundamental Motor Concept

Motors convert electrical energy into mechanical energy using electromagnetic principles. The method of energy conversion is fundamentally the same in all electric motors.