JEE Advanced 1985
LEVELJEE Advanced
Visualized Solution
The Sigma Insight: Faraday's Laws of Electromagnetic Induction
Analyzing the Setup
Imagine a semicircular conducting loop resting peacefully in a field-free region. Right next to it, separated by a sharp boundary, lies a region filled with a uniform magnetic field pointing directly into the screen.
Now, we set this loop into motion. We start rotating it with a constant angular velocity about an axis passing through its center .
As the loop spins, it begins to cross the boundary and enter the magnetic field. This is where the physics comes alive!
The Master Equation
Faraday's Law
Because the loop is entering the magnetic field, the area of the loop exposed to the field is continuously increasing. This means the magnetic flux passing through the loop is changing.
And what happens when magnetic flux changes? Faraday's Law of Induction tells us that an electromotive force (EMF) is induced!
Let's calculate exactly how much area is inside the field at any given time . The loop rotates by an angle , which is simply .
The area of this circular sector is given by the geometric formula:
Since the magnetic field is uniform and perpendicular to the plane of the loop, the magnetic flux is just the product of the magnetic field and this area:
Finding the Induced Current
To find the induced EMF, we take the time derivative of the magnetic flux. Since , , and are all constants, the derivative of is just .
This gives us a beautifully constant induced EMF:
With the EMF calculated, finding the current is straightforward. We simply divide the EMF by the loop's effective resistance .
This gives us the magnitude of the induced current, answering the first part of our problem:
Lenz's Law and the Direction of Current
Now, let's tackle the direction of the current. Look closely at the loop as it enters the magnetic field. The number of field lines pointing into the screen through the loop is increasing.
Lenz's Law states that the induced current will always fight the change that created it. To oppose the increasing inward flux, the loop needs to create its own magnetic field pointing out of the screen.
Using the right-hand grip rule, if you point your thumb out of the screen, your fingers will naturally curl in an anti-clockwise direction. Therefore, the induced current flows anti-clockwise!
Plotting the Journey
The Current-Time Graph
Finally, we need to plot the current over time for two full periods. Let's break down a single rotation. One complete spin takes a time period .
During the first half rotation (from to ), the loop is entering the field. The current is constant and anti-clockwise. If we define anti-clockwise as positive, we draw a positive horizontal line at .
In the next half rotation (from to ), the loop is exiting the magnetic field. The inward flux is now decreasing.
To oppose this decrease, the induced current flows clockwise to create more inward field. This means the current flips to negative, but maintains the exact same magnitude: .
This cycle repeats flawlessly for every rotation. Entering the field gives a positive current, and exiting gives a negative current. The result is a perfect, alternating square wave graph. And that completes our comprehensive analysis!
Similar Questions
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