A bar magnet is passing through a conducting loop of radius R with velocity v. The radius of the bar

Q: Why does the graph start with a negative peak instead of a positive one?

It depends on our sign convention for the area vector. If we take the leftward direction as positive, the flux is positive and increasing as the magnet approaches, making $\frac{d\Phi}{dt} > 0$. Faraday's law ($e = -\frac{d\Phi}{dt}$) then gives a negative emf. The key is that the entering and exiting peaks must have opposite signs.

Q: Why is the emf exactly zero when the magnet is inside?

When the magnet is fully inside, the number of magnetic field lines passing through the loop remains constant as it moves. Since the flux isn't changing ($\frac{d\Phi}{dt} = 0$), no emf is induced.

Q: Are the two peaks identical in shape?

Yes, assuming the magnet moves at a constant velocity, the rate of flux change during entry is exactly the opposite of the rate during exit, leading to symmetric but inverted peaks.

Q: What if the magnet was flipped so the South pole was on the left and North on the right?

The entire graph would flip! The entering phase would produce a positive peak, and the exiting phase would produce a negative peak.

Q: Why is there a gap labeled $l/v$ between the peaks?

The gap represents the time during which the magnet is completely inside the loop. Since the magnet has length $l$ and travels at velocity $v$, the time it spends fully inside is proportional to $l/v$. If the magnet was very short (like a point dipole), the gap would disappear.

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 emf in the loop can be represented by the approximate curve

Sigma Percentile · Accepted Answer

Imagine a conducting loop, and a bar magnet getting ready to pass right through it. We need to figure out how the induced electromotive force (emf) changes over time. This is a classic application of Faraday's and Lenz's Laws, and it requires us to carefully track the magnetic flux. Let's break this journey down into three distinct phases: entering, moving inside, and exiting.

### The Setup and Magnetic Field

First, let's look at the magnet itself. It has a North pole on the left and a South pole on the right. We know that outside a magnet, magnetic field lines travel from the North pole to the South pole. Therefore, if you look at the space along the axis of the magnet (both in front of it and behind it), the magnetic field lines actually point towards the left. Keep this leftward field in mind, as it is the key to determining the sign of our flux!

### Phase 1: The Approach

As the magnet moves to the right with velocity $v$, its South pole approaches the conducting loop. The leftward magnetic field passing through the loop gets stronger and stronger as the magnet gets closer.

If we define our area vector $\vec{A}$ to also point to the left, the magnetic flux $\Phi = \vec{B} \cdot \vec{A}$ is positive. Because the field is getting stronger, this positive flux is increasing.

According to Faraday's Law, an increasing flux induces an emf. But Lenz's Law adds a crucial minus sign—the induced emf opposes the change! The formula is:

$$e = -\frac{d\Phi}{dt}$$

Since the flux is increasing, the rate of change $\frac{d\Phi}{dt}$ is positive. Multiplying by the negative sign makes the induced emf $e$ negative. So, as the magnet enters, we get a **negative peak** on our graph.

### Phase 2: The Calm Inside

Now, the magnet is completely inside the loop. Because the magnet is uniform and its length $l$ is greater than the thickness of the loop, as it moves through this middle section, the total magnetic flux linked with the loop stays practically constant.

When the flux is constant, there is no change happening. The rate of change is zero:

$$\frac{d\Phi}{dt} = 0$$

Therefore, the induced emf drops to zero ($e = 0$). The graph stays flat along the time axis for the duration it takes the magnet to travel through, which is roughly $l/v$.

### Phase 3: The Departure

Finally, the magnet starts to exit the loop. The North pole is now moving away. The leftward magnetic field at the loop is getting weaker, which means the magnetic flux $\Phi$ is decreasing.

A decreasing flux means the rate of change $\frac{d\Phi}{dt}$ is negative. Thanks to that minus sign in Faraday's Law, a negative rate of change gives us a positive induced emf:

$$e = -(-	ext{value}) = 	ext{Positive}$$

The graph shoots up, creating a **positive peak**.

Putting it all together, we get a negative peak, a flat zero region, and then a positive peak. This perfectly matches the curve shown in option (c). The physics of electromagnetic induction is beautifully consistent!

Animated Solution for Physics - Electromagnetic Induction: 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 emf in the loop can be represented by the approximate curve

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Visualized Solution

Visualizing the Setup

Magnetic Field Direction

Phase 1: Entering the Loop

Induced EMF during Entry

Phase 2: Completely Inside

Zero EMF

Phase 3: Exiting the Loop

Induced EMF during Exit

Final Conclusion

The Sigma Insight: Faraday's Laws of Electromagnetic Induction

The Setup and Magnetic Field

Phase 1

Phase 2

Phase 3

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