Question

A bar magnet is passing through a conducting loop of radius R with velocity v. The radius of the bar magnet is such that it just passes through the loop. The induced e.m.f. in the loop can be represented by the approximate curve : 

• A. A
• B. B
• C. C
• D. D

Hint:

In problems involving Lenz's law, always remember that the induced effect opposes the *change* that causes it. This opposition manifests as a force (repulsive on approach, attractive on leaving) that slows the magnet down. A slower magnet means a smaller \(|d\Phi/dt|\), and thus a smaller induced EMF.

Answer 1

The Correct Option is A

Step 1: Understanding the Question:
We need to determine the shape of the induced EMF vs. time graph when a bar magnet passes through a conducting loop. This involves applying Faraday's Law and Lenz's Law.
Step 2: Key Formula or Approach:
Faraday's Law of Induction: \(\epsilon = -\frac{d\Phi_B}{dt}\), where \(\epsilon\) is the induced EMF and \(\Phi_B\) is the magnetic flux.
Lenz's Law: The direction of the induced current (and hence the polarity of the EMF) is such that it opposes the change in magnetic flux that produced it.
Step 3: Detailed Explanation:
Let's analyze the process in two parts:
Part 1: Magnet entering the loop.
1. As the North pole of the magnet approaches the loop, the magnetic flux through the loop (directed into the page, let's say) increases.
2. According to Faraday's law, since the flux is changing, an EMF is induced.
3. According to Lenz's Law, the induced current will create a magnetic field that opposes this increase. To do this, a North pole must be induced on the face of the loop facing the magnet. This creates a repulsive force, opposing the magnet's motion.
4. The rate of change of flux, \(d\Phi_B/dt\), is initially zero, increases to a maximum, and then decreases as the magnet's center reaches the loop. This creates an EMF pulse. Let's define the EMF induced in this part as having a negative polarity.
Part 2: Magnet leaving the loop.
1. As the magnet moves away, the South pole is leaving the loop. The magnetic flux through the loop is still in the same direction but its magnitude is now decreasing.
2. Since the flux is decreasing, \(d\Phi_B/dt\) is negative. According to Faraday's Law (\(\epsilon = -d\Phi_B/dt\)), the induced EMF will be positive. So, the second pulse must have the opposite polarity to the first.
3. According to Lenz's Law, the induced current will create a magnetic field to oppose this decrease (i.e., to support the existing flux). This means a South pole will be induced on the face of the loop facing the magnet. This creates an attractive force, again opposing the magnet's motion.
Effect on Velocity and EMF Magnitude:
1. As the magnet enters, the repulsive force slows it down.
2. As the magnet leaves, the attractive force also slows it down.
3. Therefore, the magnet is moving slower as it leaves the loop than when it entered.
4. The magnitude of the induced EMF is proportional to the speed of the magnet (\(\epsilon \propto v\)) because a higher speed leads to a faster rate of change of flux.
5. Since the magnet is moving slower when leaving, the magnitude of the second EMF pulse will be smaller than the magnitude of the first EMF pulse. The duration of the second pulse will be longer.
Step 4: Final Answer:
The correct graph must show two pulses of opposite polarity. The first pulse (e.g., negative) should be followed by a second pulse (positive). The magnitude of the second pulse should be smaller than the first. Graph 86435168223 correctly depicts this behavior.