19. The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along it

Q: Why is the emf zero when the magnet is exactly at the center of the coil?

When the magnet is at the center, the magnetic flux passing through the coil reaches its maximum value. At a maximum, the rate of change of flux ($d\phi/dt$) is momentarily zero. Since emf is proportional to this rate of change, the emf is zero.

Q: Why are the positive and negative peaks of the emf graph symmetric?

The problem specifies that the magnet moves with a 'constant velocity'. Because the speed is the same during the approach and the departure, the rate of change of flux is symmetric, leading to identical (but inverted) peaks. If the magnet were falling under gravity and accelerating, the second peak would be taller and narrower!

Q: What would the graph look like if the magnet moved faster?

If the magnet moved faster, the rate of change of flux would be higher. This means the peaks of the emf would be taller (higher maximum emf) and the entire event would happen in a shorter amount of time, making the graph narrower.

Q: Does it matter which pole of the magnet faces the coil?

Yes! If you flipped the magnet so the South pole approached first, the initial change in flux would induce a current in the opposite direction. The graph would start with a negative peak and then go to a positive peak. The key takeaway is still that the polarity flips!

Q: Why doesn't the emf just stay constant while the magnet is moving?

Emf depends on the *rate of change* of flux, not just the flux itself. As the magnet gets closer, the magnetic field gets stronger non-linearly, so the rate of change is constantly varying. This creates the smooth, wave-like curves instead of flat, constant lines.

Question

The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as

Sigma Percentile · Accepted Answer

Have you ever wondered how electricity is generated in power plants? It all comes down to a simple, elegant dance between magnets and coils of wire. This phenomenon, discovered by Michael Faraday, is the cornerstone of modern electricity. Let's dive into a classic problem that tests our understanding of this very concept.

### The Setup: A Magnet on the Move

Imagine you are standing inside a copper coil. Suddenly, a short bar magnet starts moving towards you at a constant velocity. What happens?

As the magnet moves, it carries its magnetic field with it. The number of magnetic field lines passing through the coil—what physicists call the magnetic flux, $\phi$—begins to change.

According to Faraday's Law of Induction, this changing magnetic flux induces an electromotive force (emf) in the coil. The equation is beautifully simple: 
$$e = -\frac{d\phi}{dt}$$

But the real hero of this story is that little minus sign. It represents Lenz's Law, which tells us that nature hates change. The induced emf will always try to create a current that opposes the change in flux.

### The Approach: Fighting the Change

Let's break down the motion into phases. First, the approaching phase.

As the magnet gets closer to the coil, the magnetic flux through the coil increases. Because nature hates this increase, the coil induces a current to fight back. It creates its own magnetic field pointing in the opposite direction, trying to repel the incoming magnet.

This opposition corresponds to a specific direction of induced current, which gives us an induced emf of a certain polarity. Let's call this the "positive" polarity. As the magnet gets closer and closer, the rate of change of flux increases, causing the induced emf to rise to a positive peak.

### The Center: The Eye of the Storm

Now, the magnet enters the coil. For a brief moment, when the short magnet is exactly at the center of the coil, the magnetic flux reaches its absolute maximum.

In calculus, we know that at a maximum or minimum, the derivative is zero. Therefore, at this exact moment, the rate of change of flux, $\frac{d\phi}{dt}$, is zero.

Consequently, the induced emf drops completely to zero. This is the eye of the storm—a momentary pause in the electrical push and pull.

### The Departure: The Desperate Pull

The magnet doesn't stop; it continues moving at a constant velocity and begins to leave the coil from the other side.

Now, the magnetic flux through the coil is decreasing. Once again, Lenz's Law kicks into action. The coil doesn't want the flux to decrease! To fight this change, it induces a current in the *opposite* direction to create a magnetic field that tries to attract the fleeing magnet.

Because the induced current has reversed its direction, the polarity of the induced emf must also flip. The emf is now negative. As the magnet moves away, the rate of decrease hits a maximum, creating a negative peak in the emf, before eventually fading back to zero as the magnet disappears into the distance.

### The Final Graph: A Tale of Two Polarities

Let's put it all together. The complete graph of the induced emf versus time must tell this exact story:
1. It starts at zero.
2. It rises to a positive peak as the magnet approaches.
3. It drops to zero as the magnet crosses the center.
4. It dips to a negative peak as the magnet leaves.
5. It returns to zero as the magnet moves far away.

Looking at our options, we need a graph that shows a clear reversal of polarity—a wave that crosses the horizontal time axis.

Graph (a) shows the emf staying entirely on one side of the axis, which would mean the current never reversed direction. Graph (c) only shows a single pulse, and graph (d) is just a triangle.

Only **graph (b)** perfectly captures the physical reality of the situation. It shows the emf starting positive, crossing through zero, and then going negative. This beautiful sine-like wave is the exact graphical representation of Faraday's and Lenz's Laws in action!

Animated Solution for Physics - Electromagnetic Induction: The variation of induced emf (e) with time (t) in a coil if a short bar magnet is moved along its axis with a constant velocity is best represented as

Select Answer:

Visualized Solution

Faraday's and Lenz's Laws

Approaching Phase: Flux Increases

First Half of the Graph

Leaving Phase: Flux Decreases

Second Half of the Graph

Conclusion

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

The Setup

The Approach

The Center

The Departure

The Final Graph

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