Faraday's Law of Electromagnetic Induction

Faraday's law of electromagnetic induction is a fundamental principle that describes how a changing magnetic field creates an electric current in a conductor. This phenomenon is crucial for understanding how generators, transformers, and many other electrical devices work. When a conductor, such as a coil, is exposed to a changing magnetic field, a voltage (also known as electromotive force or EMF) is induced across its ends.

The question asks about the voltage v induced in a coil according to Faraday's law, specifically considering a coil with N turns and magnetic flux φ.

Magnetic Flux and Induced Voltage

Magnetic flux (φ) is a measure of the total magnetic field passing through a given area. It is represented by the number of magnetic field lines passing through that area. For a voltage to be induced in a coil, there must be a change in this magnetic flux over time.

Faraday's law states that the magnitude of the induced voltage is directly proportional to the rate of change of magnetic flux. Mathematically, for a single loop or turn, the induced voltage v is given by:

\($$v = - \frac{d{\phi}}{dt}$$\)

The negative sign indicates the direction of the induced voltage, as described by Lenz's Law, which states that the induced current will flow in a direction that opposes the change in magnetic flux that caused it. However, for calculating the magnitude of the voltage, we often consider the absolute value, focusing on the strength of the induced voltage.

Coil Turns and Total Induced Voltage

When a coil has multiple turns, say N turns, the induced voltage in each turn adds up. Therefore, the total voltage v induced across a coil with N turns is N times the voltage induced in a single turn. This means the induced voltage is directly proportional to the number of turns in the coil.

Combining the effect of the rate of change of magnetic flux and the number of turns N, the formula for the voltage v induced in a coil with N turns and magnetic flux φ is given by:

\($$v = N \frac{d{\phi}}{dt}$$\)

This formula is central to understanding how voltage is generated in many electrical systems. It highlights that a larger number of turns or a faster rate of change of magnetic flux will result in a greater induced voltage.

Analyzing the Options

Let's examine the provided options based on our understanding of Faraday's law and the relationship between induced voltage, number of turns, and magnetic flux:

• Option 1: \(v=\dfrac{1}{N}\dfrac{d{\phi}}{dt}\) — This option suggests the voltage induced is inversely proportional to the number of turns N, which contradicts Faraday's law. According to the law, more turns lead to more induced voltage.
• Option 2: \(v= {N^2}\frac{d{\phi}}{dt}\) — This option suggests the voltage induced is proportional to the square of the number of turns N, which is not the relationship described by Faraday's law.
• Option 3: \(v=N\dfrac{d{\phi}}{dt}\) — This option correctly represents Faraday's law for a coil with N turns. It shows that the induced voltage v is directly proportional to the number of turns N and the rate of change of magnetic flux \(\dfrac{d{\phi}}{dt}\).
• Option 4: \(v=N\dfrac{d{^2\phi}}{dt^2}\) — This option involves the second derivative of magnetic flux, \(\dfrac{d{^2\phi}}{dt^2}\), implying a relationship with the rate of change of the rate of change of flux. Faraday's law for induced voltage is concerned with the first derivative, i.e., the rate of change of magnetic flux itself.

Conclusion: The Correct Faraday's Law Formula

Based on the principles of electromagnetic induction, specifically Faraday's law, the voltage v induced in a coil with N turns is directly proportional to the number of turns and the rate at which the magnetic flux (φ) through the coil changes over time.

Therefore, the correct mathematical expression for the induced voltage v is:

\($$v = N \frac{d{\phi}}{dt}$$\)