Electromagnetic Induction

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Textbook

• University Physics Volume 2: Chapter 13 - Electromagnetic Induction

Magnetic Flux

Magnetic flux $\Phi$ through a surface is the product of the magnetic field $B$ and the area $A$ perpendicular to the field:

$\Phi =B\cdot A\cdot \cos \theta$

Where:

• $B$ is the magnetic field strength,
• $A$ is the area through which the field passes,
• $\theta$ is the angle between the magnetic field and the normal to the surface.

The unit of magnetic flux is the Weber (Wb).

Faraday's and Lenz's Law

Faraday's Law: The induced electromotive force (${\mathcal {E}}$) in a closed loop is equal to the negative rate of change of magnetic flux through the loop:

${\mathcal {E}}=-{\frac {d\Phi }{dt}}$

Lenz's Law: The direction of the induced EMF opposes the change in magnetic flux, which is represented by the negative sign in Faraday's Law.

Applications

Eddy Currents - Magnetic Damping/Brakes

Eddy currents are circulating currents induced in a conductor when it experiences a changing magnetic field. The induced currents produce a magnetic field that opposes the original change, creating a damping force.

Generator

In an electrical generator, mechanical energy is converted to electrical energy by rotating a coil in a magnetic field. The EMF induced in the coil varies sinusoidally over time.:

${\mathcal {EMF}}=NBA\omega \sin(\omega t)$

Where:

• $N$ is the number of turns in the coil,
• $B$ is the magnetic field strength,
• $A$ is the area of the coil,
• $\omega$ is the angular velocity of the coil’s rotation.

Back EMF of an Electric Motor

When an electric motor operates, a “back EMF” is generated, opposing the input voltage. The back EMF ${\mathcal {EMF}}_{\text{back}}$ is proportional to the speed of the motor’s rotation:

${\mathcal {EMF}}_{\text{back}}=NBA\omega$

This reduces the effective voltage driving the motor as it speeds up, limiting its maximum current.

Transformer

A transformer changes the voltage in an alternating current (AC) circuit by means of electromagnetic induction between two coils, called the primary and secondary. The relationship between the primary and secondary voltages and the number of turns is given by:

${\frac {V_{s}}{V_{p}}}={\frac {N_{s}}{N_{p}}}$

Where:

• $V_{s}$ and $V_{p}$ are the secondary and primary voltages, respectively,
• $N_{s}$ and $N_{p}$ are the number of turns in the secondary and primary coils, respectively.

This relationship enables voltage step-up or step-down based on the ratio of turns.

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