The arm PQ of a rectangular conductor is moving from x = 0 to x = 2b outwards and then inwards from

Q: Why doesn't the flux keep increasing when the arm moves from $x=b$ to $x=2b$?

The magnetic field only exists between $x=0$ and $x=b$. Once the arm passes $x=b$, it is enclosing more empty space, but no additional magnetic field lines. Therefore, the total magnetic flux (Field $\times$ Area) maxes out and remains constant.

Q: Why is the EMF negative initially?

According to Faraday's Law, $e = -d\Phi/dt$. As the arm moves from $0$ to $b$, the flux is increasing, meaning $d\Phi/dt$ is positive. The negative sign in the formula (representing Lenz's Law) makes the induced EMF negative.

Q: Why is the EMF zero from $x=b$ to $x=2b$?

In this region, the magnetic flux is constant. The derivative of a constant is zero. Since there is no *change* in flux, no EMF is induced, regardless of how fast the arm is moving.

Q: Why does the EMF become positive on the return journey?

As the arm moves back from $x=b$ to $x=0$, the area inside the magnetic field is decreasing. This means the flux is decreasing, so $d\Phi/dt$ is negative. The negative sign in Faraday's Law cancels this out, resulting in a positive EMF.

Q: Why is the power curve always positive?

Electrical power dissipated as heat in a resistor is calculated as $P = e^2/R$. Because the EMF $e$ is squared, the result is always positive, whether the EMF itself is positive or negative. Energy is always being lost as heat, never gained.

Question

The arm PQ of a rectangular conductor is moving from $x = 0$ to $x = 2b$ outwards and then inwards from $x = 2b$ to $x = 0$ as shown in the figure. A uniform magnetic field perpendicular to the plane is acting from $x = 0$ to $x = b$. Identify the graph showing the variation of different quantities with distance.

Options:

(A) A - flux, B - power dissipated, C - emf

(B) A - power dissipated, B - flux, C - emf

(C) A - flux, B - emf , C - power dissipated

(D) A - emf , B - power dissipated, C - flux

Sigma Percentile · Accepted Answer

Imagine you are sliding a conducting arm $PQ$ along a set of parallel rails. It starts at the origin ($x=0$), travels all the way to $x=2b$, and then makes a return trip back to the start.

But there is a catch—the magnetic field doesn't exist everywhere. It is strictly confined to the region between $x=0$ and $x=b$. Let's break down how this spatial constraint affects the magnetic flux, the induced EMF, and the power dissipated, and match them to the given graphs.

### Phase 1: The Journey of Magnetic Flux (Curve A)

First, let's track the magnetic flux, denoted by $\Phi$. We know flux is the product of the magnetic field and the area enclosed by the loop: $\Phi = B \cdot A$.

As the arm moves from $x=0$ to $x=b$, the area of the loop *inside* the magnetic field increases linearly. Therefore, the magnetic flux also increases linearly.

Now, what happens when the arm crosses $x=b$? It enters a region with zero magnetic field. The physical area of the loop keeps increasing, but the area *inside* the magnetic field is now maxed out at $l \cdot b$ and remains constant. So, the flux remains constant all the way to $x=2b$, and also on the return trip back to $x=b$.

Finally, as it returns from $x=b$ to $x=0$, the enclosed area shrinks, and the flux drops back to zero linearly. If you look at the graph, **Curve A** perfectly matches this behavior—a linear rise, a flat plateau, and a linear fall.

### Phase 2: The Derivative Dance - Induced EMF (Curve B)

Next, let's figure out the induced EMF. According to Faraday's Law of electromagnetic induction, the induced EMF is the negative rate of change of magnetic flux: $e = -\frac{d\Phi}{dt}$.

Mathematically, the EMF is the negative slope of our flux graph. Let's apply this to our three distinct regions.

From $x=0$ to $x=b$, the flux increases at a constant rate, meaning its slope is positive. Because of the negative sign in Faraday's law, the EMF becomes a constant negative value ($-Blv$).

From $x=b$ to $x=2b$ and back, the flux is constant, so its slope is zero, making the EMF zero.

On the final return from $x=b$ to $x=0$, the flux decreases, giving a negative slope. The double negative makes the EMF a constant positive value ($+Blv$). This step-like behavior—negative, zero, positive—is exactly what **Curve B** shows.

### Phase 3: The Energy Toll - Power Dissipation (Curve C)

Lastly, let's analyze the electrical power dissipated as heat in the resistance of the loop. The formula for power is $P = \frac{e^2}{R}$.

Notice the square term? This is crucial. It means that regardless of whether the EMF is positive or negative, the power dissipated will always be a **positive quantity**. You can never have negative power dissipation here.

So, when the arm moves from $x=0$ to $x=b$, the EMF is negative, but squaring it makes the power positive. In the field-free region where EMF is zero, the power is naturally zero. And on the return journey from $x=b$ to $x=0$, the EMF is positive, so the power is again positive.

This gives us two positive rectangular pulses, which perfectly matches **Curve C**.

### Conclusion

Putting it all together, we have successfully decoded the graphs. Curve A is the magnetic flux, Curve B is the induced EMF, and Curve C is the power dissipated. This beautifully connects Faraday's law, Lenz's law, and Joule heating into a single visual story.

Select Answer:

Visualized Solution

$Understanding the Setup$

$Magnetic Flux (Φ)$

$Flux in the Field-Free Region$

$Induced EMF (e)$

$Tracing the EMF Curve$

$Power Dissipated (P)$

$Tracing the Power Curve$

$Final Conclusion$

The Sigma Insight: Motional EMF

Phase 1

Phase 2

Phase 3

Conclusion

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The magnetic field in a region is given by $B = B_{0} (\frac{x}{a}) \hat{k}$ . A square loop of side $d$ is placed with its edges along the X and Y-axes. The loop is moved with a constant velocity $v = v_{0} \hat{i}$ . The emf induced in the loop is

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Visualized Solution

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Understanding the Setup

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Magnetic Flux (Φ)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Flux in the Field-Free Region

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Induced EMF (e)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tracing the EMF Curve

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Power Dissipated (P)

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Tracing the Power Curve

.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Final Conclusion