Animated Solution for Physics - Electromagnetic Induction: At time $t=0$ magnetic field of 1000 gauss is passing perpendicularly through the area defined by the closed loop shown in the figure. If the magnetic field reduces linearly to 500 gauss, in the next 5 s, then induced emf in the loop is

Imagine a closed conducting loop resting peacefully in a uniform magnetic field. The field lines are piercing perpendicularly through the plane of the loop. Initially, at $t=0$, the magnetic field strength is a robust $1000\text{ Gauss}$. However, this field isn't static; it begins to weaken linearly, dropping to $500\text{ Gauss}$ over the next $5\text{ seconds}$.

Our mission is to determine the electromotive force (emf) induced in this loop as a result of the changing magnetic environment. To do this, we must first understand the exact geometry of the loop to calculate its area.

At first glance, the loop's shape might seem a bit irregular. But let's break it down into simpler, more familiar geometric figures. We can visualize the entire shape as a large rectangle with dimensions $16\text{ cm}$ by $4\text{ cm}$.

However, the bottom corners of this rectangle are missing. Specifically, two right-angled triangles have been cut out from the bottom left and bottom right. By carefully observing the dimensions provided in the diagram, we can deduce that each of these triangular cutouts has a base of $4\text{ cm}$ and a height of $2\text{ cm}$.

Let's calculate the effective area of our conducting loop:

Crucial Step: In physics, consistency in units is paramount. We must convert this area into standard SI units (square meters) before proceeding further.

Faraday's Law

With the area locked in, we turn to the cornerstone of electromagnetic induction: Faraday's Law. This law states that the induced emf ($E$) in a closed circuit is equal to the negative rate of change of magnetic flux ($\Phi$) through it.

Since the magnetic flux $\Phi$ is the product of the magnetic field $B$ and the perpendicular area $A$ ($\Phi =B\cdot A$), and our loop's area remains perfectly constant, we can pull the area out of the difference operator:

Now, it's time to substitute our known values into the master equation. We know the initial magnetic field $B_1=1000\text{ G}$ and the final magnetic field $B_2=500\text{ G}$. The time interval $\Delta t$ is $5\text{ s}$.

Remember to convert Gauss to Tesla ($1\text{ Gauss}=10^{-4}\text{ Tesla}$):

Let's find the rate of change of the magnetic field:

Finally, we plug this rate back into our emf equation along with the area:

Notice how the negative signs beautifully cancel each other out, leaving us with a positive magnitude for the induced emf:

This elegant result perfectly matches option (c). As a bonus thought, according to Lenz's Law, since the outward magnetic field is decreasing, the induced current will flow in an anti-clockwise direction to generate its own outward magnetic field, attempting to oppose the change and maintain the status quo.