Animated Solution for Physics - Electromagnetic Induction: In the given figure, the magnetic flux through the loop increases according to the relation ${\varphi }_B(t)=10t^2+20t$, where ${\varphi }_B$ is in milliwebers and $t$ is in seconds. The magnitude of current through $R=2\Omega$ resistor at $t=5$ s is ...... mA.

Imagine you are observing a closed rectangular conducting loop placed in a region of space. This isn't just any empty space; it's permeated by a magnetic field that is pointing directly into the plane of your screen. You can visualize this field as a series of arrows flying away from you, piercing through the area enclosed by the loop.

Now, the universe rarely stays static. The problem tells us that the magnetic flux—the total number of these magnetic field lines passing through the loop—is not constant. It is increasing with time according to a specific mathematical relation: ${\varphi }_B(t)=10t^2+20t$. Here, the flux is measured in milliwebers (mWb), and time $t$ is in seconds.

Our goal is to find the exact magnitude of the electric current flowing through the $2\Omega$ resistor attached to this loop at the precise moment when $t=5$ seconds.

Faraday's Law

When a magnetic field passing through a conducting loop changes, nature reacts. It doesn't like this change. According to Faraday's Law of Electromagnetic Induction, this changing magnetic flux induces an electromotive force (EMF) in the loop. You can think of this EMF as an electrical pressure that pushes charges around, creating a current.

The magnitude of this induced EMF is directly proportional to the rate at which the magnetic flux is changing. Mathematically, this is expressed as the time derivative of the flux:

This is our master key. To find the EMF at any given time, we need to differentiate our flux equation with respect to time.

Let's bring in our flux equation and apply the derivative. We substitute ${\varphi }_B(t)=10t^2+20t$ into Faraday's Law:

Now, we perform the atomic compute. The derivative of $10t^2$ is $20t$, and the derivative of $20t$ is simply $20$.

We must be very careful with our units here. Since the original flux was given in milliwebers (mWb), the resulting induced EMF will be in millivolts (mV). So, our time-dependent EMF equation is $|E|=(20t+20)$ mV.

Ohm's Law

We have the electrical pressure (EMF), but the question asks for the current. This is where Ohm's Law bridges the gap. Ohm's Law states that the current $i$ is equal to the voltage (or EMF) divided by the resistance $R$:

We know our loop has a resistance of $R=2\Omega$. Let's substitute our EMF expression and the resistance into Ohm's Law:

Simplifying this fraction by dividing both terms in the numerator by 2, we get a beautiful, clean linear equation for the current:

Because our EMF was in millivolts and our resistance in ohms, our current is naturally in milliamperes (mA).

We are now in the endgame. We have the formula for the current at any time $t$, and we need to find its value exactly at $t=5$ seconds.

Let's substitute $t=5$ into our current equation:

Executing the final arithmetic: $10\times 5=50$, and $50+10=60$.

$$i=60\text{ mA}$$

At exactly 5 seconds, a current of 60 mA is flowing through the resistor.

Lenz's Law

While the question only asked for the magnitude, true mastery requires understanding the direction. This is governed by Lenz's Law, which states that the induced current will flow in a direction that opposes the change in flux that created it.

Since the magnetic flux pointing into the page is increasing, the loop wants to fight this increase. It does so by generating its own magnetic field pointing out of the page. Using the right-hand grip rule, to create an outward-pointing magnetic field, the induced current must flow in an anti-clockwise direction around the loop.

Always keep Lenz's Law in your toolkit; JEE loves to test your conceptual depth by asking for both magnitude and direction!