When a loop of current-carrying wire turns continuously in a magnetic field, a ___ is created

The turning loop generates a change in magnetic flux through the area enclosed by the loop, which according to Faraday's Law of Electromagnetic Induction, induces an emf. This is mathematically represented as:

Where $\mathcal{E}$ is the electromotive force (emf) and $\Phi_B$ denotes the magnetic flux.

If the number of turns in the wire loop is $N$, the induced emf becomes:

This can also be viewed in the context of motional emf, which occurs if part of a circuit moves through a magnetic field. The magnitude of the motional emf can be described by:

Where $B$ is the magnetic field strength, $l$ is the length of the wire in the magnetic field, $v$ is the velocity of the wire, and $\theta$ is the angle between the velocity and the magnetic field direction.

The continuous rotation of the loop results in a periodic change in the magnetic flux and thus generates an alternating current (AC) if the loop is part of a complete circuit. This forms the basis of many electric generators and induction motors.

Explanation

Faraday's law of electromagnetic induction describes how a magnetic field can interact with an electric circuit to produce an electromotive force (EMF). This is a fundamental principle of electromagnetism, and it forms the basis for many electrical devices such as transformers, electric generators, and inductors.

Faraday's Law

Faraday's law can be mathematically expressed as:

Here, $\mathcal{E}$ is the EMF (electromotive force) generated, and $\Phi_B$ is the magnetic flux. The negative sign indicates the direction of the induced EMF and current based on Lenz's Law, which states that the induced EMF will oppose the change in magnetic flux.

Magnetic Flux

Magnetic flux, $\Phi_B$, is given by:

Where:

• $\mathbf{B}$ is the magnetic field,
• $d\mathbf{A}$ is a differential area element,
• $\cdot$ denotes the dot product, indicating that only the component of the magnetic field perpendicular to the area contributes to the flux.

Lenz's Law

Lenz's Law is incorporated into Faraday's law to determine the direction of the induced EMF. It is represented by the negative sign in the formula and ensures that the induced EMF creates a magnetic field that opposes the change in the original magnetic field.

Applications

Faraday's law of electromagnetic induction is crucial in designing and operating numerous electrical devices:

• Transformers: Use electromagnetic induction to transfer electrical energy between circuits.
• Generators: Convert mechanical energy into electrical energy.
• Inductors: Store energy in a magnetic field when electrical current passes through them.

This law illustrates the interrelationship between time-varying magnetic fields and electric fields, emphasizing the fundamental nature of electromagnetism in physics and engineering.

Understanding Motional Electromotive Force (emf)

Motional Electromotive Force (emf) arises when a conductor moves through a magnetic field, causing the free charges within the conductor to experience a force. This motion results in a potential difference (voltage) across the conductor. The concept is deeply rooted in Faraday's Law of Induction and Lorentz Force.

Faraday's Law of Induction

This law states that a changing magnetic environment of a coil of wire induces an electromotive force (emf) in the coil. For a conductor moving with a velocity $\mathbf{v}$ through a magnetic field $\mathbf{B}$, the induced emf can be calculated using:

Where $\Phi$ is the magnetic flux.

Lorentz Force

The Lorentz Force is the force experienced by a charge q moving with a velocity $\mathbf{v}$ through a magnetic field $\mathbf{B}$:

Derivation of Motional emf

Consider a conductor of length $L$ moving with a velocity $\mathbf{v}$ perpendicular to a uniform magnetic field $\mathbf{B}$. The induced emf ( $\mathcal{E}$ ) can be expressed as:

For the case where the conductor's motion is perpendicular to the magnetic field ($\theta = 90^\circ$):

Here:

• $B$ is the magnetic field strength.
• $L$ is the length of the conductor.
• $v$ is the velocity of the conductor.

Applications

Motional emf is a fundamental principle in:

• Electric Generators: Rotation of coils through magnetic fields to generate electricity.
• Railguns: Using rapidly moving conductive rails to generate large currents.
• Induction Heating: Movement of conductors in dynamic magnetic fields to generate heat.

Summary

Motional electromotive force is a crucial concept in electromagnetism, governing how mechanical motion can result in electrical potential. Understanding this concept is essential for comprehending a wide range of electrical and electronic systems.