Animated Solution for Physics - Electromagnetic Induction: A circular coil of radius $8.0\text{ cm}$ and $20$ turns is rotated about its vertical diameter with an angular speed of $50{\text{ rad s}}^{-1}$ in a uniform horizontal magnetic field of $3.0\times 10^{-2}\text{ T}$. The maximum emf induced in the coil will be ...... $\times 10^{-2}\text{ V}$. (rounded off to the nearest integer.)

Imagine you are standing in a laboratory, looking at a beautifully simple yet profoundly powerful setup: a circular coil of wire suspended in mid-air, spinning rapidly like a coin on a table. But this isn't just any spin; it's rotating precisely about its vertical diameter. Surrounding this spinning coil is an invisible, uniform horizontal magnetic field.

This is the exact physical reality of an Alternating Current (AC) generator. As the coil spins, the amount of magnetic field lines piercing through its surface—what we call the magnetic flux—is constantly changing. Sometimes the coil faces the field head-on, capturing maximum flux, and a split second later, it's edge-on, capturing none. This continuous, rhythmic change in flux is the heartbeat of electromagnetic induction.

Faraday's Law

To understand how this spinning motion creates electricity, we must invoke one of the most elegant laws in all of physics: Faraday's Law of Electromagnetic Induction. Faraday discovered that nature abhors a change in magnetic flux, and it responds by inducing an electromotive force (EMF) to oppose that change.

Mathematically, the magnetic flux $\Phi$ through a coil of $N$ turns, area $A$, in a magnetic field $B$ is given by the dot product of the magnetic field vector and the area vector:

Because the coil is rotating with a constant angular velocity $\omega$, the angle $\theta$ between the area vector and the magnetic field changes with time as $\theta =\omega t$. Substituting this in, we get:

Now, Faraday's Law tells us that the induced EMF $\epsilon$ is the negative rate of change of this flux:

Let's take the derivative of our flux equation with respect to time. The derivative of $\cos (\omega t)$ is $-\omega \sin (\omega t)$. The negative signs cancel out, leaving us with a beautiful, oscillating function:

The question specifically asks for the maximum EMF induced in the coil. Looking at our equation, the sine function oscillates between $-1$ and $1$. Therefore, the maximum possible value of this EMF—the amplitude of our AC signal—is simply the coefficient in front of the sine term:

This is our master equation. It tells us that to get a massive voltage, we need a strong magnet ($B$), a large coil ($A$), lots of turns ($N$), and we need to spin it incredibly fast ($\omega$).

Raw Substitution

Now that we have our master equation, let's gather the raw data provided in the problem. We are given: - Number of turns, $N=20$ - Magnetic field, $B=3.0\times 10^{-2}\text{ T}$ - Angular speed, $\omega =50{\text{ rad s}}^{-1}$ - Radius of the coil, $r=8.0\text{ cm}$

Before we plug anything in, we must be vigilant about units. Physics demands consistency. The radius is in centimeters, which is a trap! We must convert it to standard SI units (meters):

Now, we can calculate the area $A$ of the circular coil:

Let's substitute all these raw values into our maximum EMF formula without evaluating them just yet. Let's see the raw structure of the physics:

Simplifying the Math

I know this long string of numbers might look intimidating, but let's take a breath. The art of calculation in physics is all about grouping friendly numbers together.

First, let's look at $20$ and $50$. Multiplying them is incredibly satisfying:

Next, let's handle the area. Squaring $0.08$ gives us $0.0064$. So the area is:

Now, let's rewrite our expression with these simplified chunks:

Look at how much cleaner that is! Now, let's combine the powers of ten. We have $1000$ (which is $10^3$) multiplied by $10^{-2}$. This simply leaves us with $10$.

Multiply that $10$ by the $3.0$ from the magnetic field, and we get a neat $30$:

Our massive equation has now collapsed into a very manageable product:

Multiplying $30$ by $0.0064$ gives us $0.192$. So, we are left with:

Rounding to Victory

We are at the finish line. We need a numerical answer, so let's substitute the approximate value of $\pi \approx 3.14159$:

The problem asks us to format our answer as something multiplied by $10^{-2}\text{ V}$. To do this, we shift the decimal point two places to the right:

Finally, the question demands that we round off to the nearest integer. Looking at $60.318$, the decimal part is less than $0.5$, so we round down.

Changing the Axis

Before we close this chapter, let's do a quick thought experiment. What if the coil was rotated about a horizontal axis that was perfectly parallel to the magnetic field?

Visualize it. The coil is spinning, but its face is always parallel to the magnetic field lines. The area vector $\vec{A}$ (which points straight out of the face of the coil) would always be perpendicular to the magnetic field $\vec{B}$.

Because the dot product of two perpendicular vectors is zero, the magnetic flux $\Phi$ would be permanently zero. If the flux never changes, Faraday's Law tells us that absolutely no EMF would be induced!

This highlights a profound truth in electromagnetism: it's not just about having a magnetic field and a moving coil; the relative orientation is everything. The geometry of the setup dictates the physics.