D.4.1 Principles of electromagnetic induction (HL only)

Magnetic Flux and Electromagnetic Induction

Magnetic Flux: A Measure of Magnetic Field Through a Surface

It is defined as:

$$\mathrm{\Phi }=BA\cos \theta$$

where:

• $B$ is the magnetic flux density (measured in teslas, T).
• $A$ is the area of the surface (measured in square meters, $m^2$).
• $\theta$ is the angle between the magnetic field lines and the normal (perpendicular) to the surface.

Understanding the Formula

1. When the field is perpendicular to the surface $(\theta =0^{\circ })$, the flux is maximized: $$\mathrm{\Phi }=BA$$
2. When the field is parallel to the surface $(\theta ={90}^{\circ })$, the flux is zero: $$\mathrm{\Phi }=0$$
3. For angles in between, the flux is reduced by the factor $\cos \theta$.

Faraday’s Law: Inducing Emf Through Changing Magnetic Flux

Mathematically, this is expressed as:

$$\epsilon =-N\frac{\mathrm{\Delta }\mathrm{\Phi }}{\mathrm{\Delta }t}$$

where:

• $\epsilon$ is the induced emf (measured in volts, V).
• $N$ is the number of turns in the coil.
• $\mathrm{\Delta }\mathrm{\Phi }$ is the change in magnetic flux (measured in webers, Wb).
• $\mathrm{\Delta }t$ is the time interval over which the change occurs (measured in seconds, s).

Factors Affecting Induced Emf

1. Rate of Change of Flux: Faster changes produce larger emf.
2. Number of Turns: More turns result in greater emf.
3. Magnitude of Flux Change: Larger changes in flux induce more emf.

Lenz’s Law: Opposing the Change

Applying Lenz’s Law

1. If the flux increases, the induced current creates a magnetic field that opposes the increase.
2. If the flux decreases, the induced current creates a magnetic field that supports the original field, opposing the decrease.

Why Does Lenz’s Law Matter?

1. Lenz’s Law ensures that energy is not created or destroyed.
2. The work done to induce the emf is converted into electrical energy, which may be dissipated as heat or used to do work.

Induced Emf in Conductors: Motional Emf

When a conductor moves through a magnetic field, an emf is induced across its ends.

This is known as motional emf and is given by:

$$\epsilon =BvL$$

where:

• $B$ is the magnetic flux density (measured in teslas, T).
• $v$ is the velocity of the conductor (measured in meters per second, $\text{m/s}$).
• $L$ is the length of the conductor (measured in meters, m).

How Motional Emf Works

1. As the conductor moves through the magnetic field, free electrons within it experience a magnetic force.
2. This force causes the electrons to accumulate at one end of the conductor, creating a potential difference (emf) across its ends.

Applications and Implications of Electromagnetic Induction

Electromagnetic induction is the foundation of many technologies, including:

• Generators: Convert mechanical energy into electrical energy by rotating a coil in a magnetic field.
• Transformers: Use changing magnetic flux to transfer energy between coils.
• Induction Cooktops: Use induced currents to heat cookware directly.

Reflection

Summary

• Magnetic flux $(\mathrm{\Phi })$ measures the amount of magnetic field passing through a surface and is given by $\mathrm{\Phi }=BA\cos \theta$.
• Faraday’s law states that a changing magnetic flux induces an emf, expressed as $\epsilon =-N\frac{\mathrm{\Delta }\mathrm{\Phi }}{\mathrm{\Delta }t}$.
• Lenz’s law ensures the induced emf opposes the change in flux, conserving energy.
• Motional emf in a moving conductor is given by $\epsilon =BvL$.