Animated Solution for Physics - Electromagnetic Induction: A circular coil of radius $10\text{ cm}$ is placed in a uniform magnetic field of $3.0\times 10^{-5}\text{ T}$ with its plane perpendicular to the field initially. It is rotated at constant angular speed about an axis along the diameter of coil and perpendicular to magnetic field, so that it undergoes half of rotation in $0.2\text{ s}$. The maximum value of emf induced (in $\mu \text{V}$) in the coil will be close to the integer ......... .

The phenomenon of electromagnetic induction is arguably one of the most profound and world-changing discoveries in the history of physics. It forms the bedrock of our modern electrical grid, bridging the gap between mechanical motion and electrical energy. In this problem, we are going to explore a classic, elegant application of Faraday's Law: a rotating circular coil placed in a uniform magnetic field. By the end of this journey, you will see exactly how the fundamental principles of electric generators work!

Let us start by painting a clear mental picture of the scenario. Imagine a perfectly circular coil made of conducting wire. The radius of this coil is given as $r=10\text{ cm}$, which we must immediately convert to standard SI units as $0.1\text{ m}$ to avoid any dimensional mismatches later on.

This coil is submerged in a uniform magnetic field. The strength of this magnetic field is $B=3.0\times 10^{-5}\text{ T}$. To give you some context, this is roughly on the same order of magnitude as the Earth's natural magnetic field!

Initially, the problem states that the plane of the coil is perpendicular to the magnetic field. This is a crucial piece of geometry. In physics, we define the orientation of a surface using an "area vector" $\vec{A}$. This vector is always perfectly perpendicular (normal) to the surface itself. Therefore, if the plane of the coil is perpendicular to the magnetic field lines, the area vector $\vec{A}$ is perfectly parallel to the magnetic field $\vec{B}$. At this exact starting moment, the angle between them is zero.

The coil doesn't just sit there; it is forced to rotate about an axis that lies along its diameter. We are told that it completes exactly half of a rotation in $0.2\text{ s}$.

In rotational kinematics, the time it takes to complete one full revolution is called the time period, denoted by $T$. Since half a rotation takes $0.2\text{ s}$, a full rotation will naturally take twice as long.

Now that we have the time period, we can determine the angular velocity $\omega$ of the coil. Angular velocity tells us how many radians the coil sweeps through every second. The relationship is straightforward:

Substituting our time period into the equation:

This means that every second, the coil rotates through $5\pi$ radians. This constant angular speed is the mechanical input that will soon drive our electrical output.

Magnetic Flux and Faraday's Law

As the coil rotates, the orientation of its area vector $\vec{A}$ changes relative to the fixed magnetic field $\vec{B}$. Consequently, the angle $\theta$ between them changes continuously. Because the rotation is uniform, we can express this angle as a simple function of time: $\theta =\omega t$.

The concept of magnetic flux, denoted by $\varphi$, is a measure of how many magnetic field lines are piercing through the area of the coil. Mathematically, it is defined as the dot product of the magnetic field vector and the area vector:

Substituting our time-dependent angle, the flux becomes a dynamic quantity:

This is where the magic happens. According to Faraday's Law of Electromagnetic Induction, nature abhors a change in magnetic flux. Whenever the magnetic flux linked with a closed loop changes over time, an electromotive force (EMF) is induced in the loop. The magnitude of this induced EMF $e$ is directly proportional to the rate of change of the flux. Including Lenz's Law (which gives us the direction), the complete equation is:

Let's apply some calculus and differentiate our flux expression with respect to time $t$:

Since $B$ and $A$ are constants, they pull out of the derivative. The derivative of $\cos (\omega t)$ is $-\omega \sin (\omega t)$. The negative signs cancel out beautifully:

The equation we just derived, $e=BA\omega \sin (\omega t)$, tells us that the induced EMF is not constant; it is an alternating voltage (AC) that varies sinusoidally with time.

The problem specifically asks for the maximum value of this induced EMF. To find this, we must look at the mathematical properties of the sine function. The term $\sin (\omega t)$ oscillates smoothly between $-1$ and $1$.

Therefore, the maximum possible value of the entire expression occurs when $\sin (\omega t)=1$. This leaves us with the amplitude of the wave:

This elegant formula reveals that to get a larger maximum voltage, you can do three things: use a stronger magnetic field ($B$), use a larger coil ($A$), or spin the coil faster ($\omega$).

Now, we have reached the final stage: substituting our known numerical values into the maximum EMF equation. Let's recall that the area of our circular coil is $A=\pi r^2$.

Plugging in the numbers:

Let's carefully group the numerical coefficients and the powers of ten to avoid any silly algebraic mistakes:

Here we encounter a classic physicist's trick. In competitive exams like JEE and NEET, it is a very standard and highly useful approximation to take ${\pi }^2\approx 10$. This is because $\pi \approx 3.14159$, and squaring it gives approximately $9.869$, which is close enough to $10$ for these types of order-of-magnitude integer questions. Applying this approximation:

The problem requests the answer in microvolts ($\mu \text{V}$). Since $10^{-6}\text{ V}$ is the exact definition of $1\mu \text{V}$, we can seamlessly write our final answer as:

The integer value we are looking for is 15. This problem is a fantastic demonstration of how abstract calculus and vector geometry come together to describe the very real, tangible process of generating electricity!