Animated Solution for Physics - Electromagnetic Induction: In an AC generator, a coil with $N$ turns, all of the same area $A$ and total resistance $R$, rotates with frequency $\omega$ in a magnetic field $B$. The maximum value of emf generated in the coil is

The phenomenon of electromagnetic induction is the beating heart of modern electricity generation. When a coil spins in a magnetic field, it doesn't just move; it weaves mechanical energy into electrical energy. Let's break down the physics of an AC generator and uncover the maximum electromotive force (EMF) it can produce.

Imagine you are standing next to a massive AC generator. Inside, there is a coil consisting of $N$ turns, each with an identical area $A$. This coil is placed in a uniform magnetic field $B$ and is forced to rotate at a constant angular frequency $\omega$.

As the coil spins, the angle between its area vector and the magnetic field lines constantly changes. This continuous rotation is the secret to generating alternating current.

To understand how much voltage is generated, we first need to look at the magnetic flux passing through the coil. Magnetic flux ($\Phi$) is essentially a measure of how many magnetic field lines pierce through the coil's area.

For a single turn of the coil, the flux is given by the dot product of the magnetic field vector and the area vector:

Since our coil has $N$ turns connected in series, the total magnetic flux is simply $N$ times the flux of a single turn:

Because the coil is rotating with an angular frequency $\omega$, the angle $\theta$ at any given time $t$ is $\theta =\omega t$. Substituting this into our equation, we get the time-dependent flux:

Unleashing the EMF

Now, we bring in the heavy artillery: Faraday's Law of Induction. This fundamental law states that the induced EMF ($e$) in a closed loop is equal to the negative rate of change of the magnetic flux through it.

Mathematically, this is expressed as:

Let's substitute our flux equation into Faraday's Law and differentiate it with respect to time. I know this calculus step might look intimidating, but let's take a breath and apply the chain rule. The derivative of $\cos (\omega t)$ is $-\omega \sin (\omega t)$.

This beautiful equation tells us that the induced EMF is not constant; it oscillates sinusoidally with time, which is exactly why we call it an Alternating Current generator!

The question asks for the maximum value of the generated EMF. To find this, we need to look at our EMF equation and find its peak value.

The sine function, $\sin (\omega t)$, oscillates between $-1$ and $+1$. Therefore, its maximum possible value is exactly $1$.

Substituting $\sin (\omega t)=1$ into our equation gives us the peak EMF:

This is our final answer! Notice how the maximum EMF depends on the number of turns, the area of the coil, the strength of the magnetic field, and how fast the coil is spinning. Interestingly, it does not depend on the resistance $R$ of the coil. The resistance would only come into play if we were asked to find the maximum induced current ($I_{\text{max}}=e_{\text{max}}/R$).