The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed i

Q: Why can't the current be in the same direction in both the inner and outer loops?

If the current flowed in the same direction, it would be a simple spiral coil. However, the options and the geometry imply a twisted figure-8 loop where the inner and outer segments oppose each other to form a single closed circuit.

Q: How do we know the outer loop dominates the current direction?

The induced EMF is proportional to the area enclosed by the loop ($\mathcal{E} = -A \frac{dB}{dt}$). Since the outer loop has a much larger area, its contribution to the magnetic flux is greater, dictating the net current direction.

Q: What does a 'cross' magnetic field mean?

The crosses ($\times$) indicate that the magnetic field lines are pointing perpendicularly into the plane of the page or screen, like the tail feathers of an arrow flying away from you.

Q: Why does the induced field point out of the page?

Because the inward magnetic field is increasing, Lenz's law states the induced current will oppose this increase by creating its own magnetic field pointing outwards (represented by dots, $\odot$).

Q: Are $I_1$ and $I_2$ equal in magnitude?

Yes, since they are part of the exact same single continuous wire loop, the current magnitude must be identical throughout the entire circuit.

Question

The figure shows certain wire segments joined together to form a coplanar loop. The loop is placed in a perpendicular magnetic field in the direction going into the plane of the figure. The magnitude of the field increases with time. $I_1$ and $I_2$ are the currents in the segments $ab$ and $cd$. Then,

Options:

(A) $I_1 > I_2$

(B) $I_1 < I_2$

(C) $I_1$ is in the direction $ba$ and $I_2$ is in the direction $cd$.

(D) $I_1$ is in the direction $ab$ and $I_2$ is in the direction $dc$.

Sigma Percentile · Accepted Answer

### The Setup: A Loop in a Changing Field

Imagine you are looking at a complex wire loop placed flat on a table. The entire region is bathed in a uniform magnetic field. The crosses ($	imes$) in the diagram are a standard physics convention indicating that the magnetic field lines are piercing directly into the plane of the screen, much like the tail feathers of an arrow flying away from you.

The problem presents a critical dynamic condition: the magnitude of this inward magnetic field is increasing with time. This changing magnetic environment is the engine that drives the entire physical phenomenon we are about to analyze.

### Lenz's Law: The Universe's Resistance to Change

Nature possesses a fundamental inertia, a deep-seated resistance to change. In electromagnetism, this is elegantly captured by **Lenz's Law**. When the magnetic flux through a closed loop changes, an electromotive force (EMF) is induced. This EMF drives a current whose own magnetic field acts to oppose the original change in flux.

Since the inward magnetic field is getting stronger, the loop "fights back" by trying to create a magnetic field pointing OUT of the screen ($\odot$). To generate an outward-pointing magnetic field, the right-hand grip rule dictates that the net induced current must flow in an **anti-clockwise** direction.

### The Tale of Two Loops: Outer vs. Inner

Here is where the geometry gets fascinating. The figure shows what appears to be two concentric squares, but the problem explicitly states it is a single "coplanar loop." For a single continuous wire to form this inner-and-outer structure, it must be twisted, much like a figure-8.

Because the outer square encloses a significantly larger area ($A_{outer} > A_{inner}$), it captures more of the changing magnetic flux. Therefore, the outer loop dominates the electromagnetic response. The net current direction is dictated by this outer loop, meaning the current in the outer perimeter will flow **anti-clockwise**.

However, because the wire is twisted to form the inner square, the current path folds back on itself. As a direct geometric consequence of this figure-8 topology, the current in the inner loop must flow in the opposite rotational sense. Thus, the current in the inner loop flows **clockwise**.

### Tracing the Currents: The Final Puzzle

Now, we simply need to trace this clockwise current through the specific segments of the inner loop to find our answer.

Let's look at the top wire of the inner loop, where points $a$ and $b$ are located. A clockwise circulation means the current travels across the top wire from left to right. Since point $a$ is positioned to the left of point $b$, the current $I_1$ flows exactly from $a$ to $b$. We say $I_1$ is in the direction **$ab$**.

Next, let's examine the bottom wire of the inner loop, home to points $c$ and $d$. Continuing our clockwise path, the current along the bottom wire must flow from right to left. Since point $c$ is on the left and point $d$ is on the right, the current $I_2$ flows from $d$ to $c$. We say $I_2$ is in the direction **$dc$**.

Combining these two deductions, we conclude that $I_1$ is in the direction $ab$ and $I_2$ is in the direction $dc$. This flawless logical sequence leads us directly to the correct option.

Select Answer:

Visualized Solution

Visual Anchor

Lenz's Law

Outer Loop Current

Inner Loop Current

Direction of $I_{1}$

Direction of $I_{2}$

Final Conclusion

The Sigma Insight: Lenz's Law

The Setup

Lenz's Law

The Tale of Two Loops

Tracing the Currents

Similar Questions

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Select Answer:

Visualized Solution

Visual Anchor

Lenz's Law

Outer Loop Current

Inner Loop Current

Direction of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } I1​

Direction of .math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } I2​

Final Conclusion

The Sigma Insight: Lenz's Law

The Setup

Lenz's Law

The Tale of Two Loops

Tracing the Currents

Similar Questions

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.math-text-wrapper .katex { font-size: 1.05em !important; } .math-text-wrapper .katex-display { margin: 1rem 0 !important; } Two identical circular loops of metal wire are lying on a table without touching each other. Loop A carries a current which increases with time. In response, the loop B

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