A metallic square loop ABCD is moving in its own plane with velocity v in a uniform magnetic field p

Q: Why is there no current flowing in the loop?

Since the loop is moving in a uniform magnetic field, the induced emf in arm $AD$ is exactly equal and opposite to the induced emf in arm $BC$. They cancel each other out in the closed loop, resulting in zero net current.

Q: Is an electric field induced in arms $AB$ and $CD$?

Yes! The formula $\vec{E} = -(\vec{v} \times \vec{B})$ applies everywhere in the moving conductor. In arms $AB$ and $CD$, this field points across the width of the wire, creating a tiny potential difference (Hall effect), but the options specifically ask about $AD$ and $BC$.

Q: What if the magnetic field was non-uniform?

If the magnetic field was non-uniform, the induced emf in $AD$ and $BC$ would be different. This would create a net emf in the loop, causing a current to flow!

Q: Why does the electric field point upwards?

The magnetic force pushes positive charges downwards (and electrons upwards). This makes the bottom of the loop positive and the top negative. Since electric field lines point from positive to negative, the induced electric field points upwards.

Q: Does this violate Faraday's Law?

Not at all! Faraday's Law says the induced emf in a closed loop equals the rate of change of magnetic flux. Here, the flux through the loop is constant, so the net emf is zero. The motional emfs in $AD$ and $BC$ perfectly cancel each other out.

Question

A metallic square loop $ABCD$ is moving in its own plane with velocity $v$ in a uniform magnetic field perpendicular to its plane as shown in the figure. Electric field is induced

Options:

(A) in $AD$, but not in $BC$

(B) in $BC$, but not in $AD$

(C) neither in $AD$ nor in $BC$

(D) in both $AD$ and $BC$

Sigma Percentile · Accepted Answer

The phenomenon of motional EMF is one of the most elegant intersections of electricity and magnetism. When a conductor moves through a magnetic field, it doesn't just sit there passively—it becomes a dynamic environment where charges are pushed, pulled, and ultimately balanced. Let's dive into the physics of a metallic square loop moving through a uniform magnetic field and uncover where and why an electric field is induced.

### The Moving Conductor

Imagine a metallic square loop, $ABCD$, gliding smoothly to the right with a constant velocity $\vec{v}$. It is passing through a uniform magnetic field $\vec{B}$ that points straight out of the screen towards you.

Inside this metallic loop, there is a sea of free electrons. As the loop moves, these electrons are carried along with it. They are moving charges in a magnetic field, which means they are about to experience a fundamental force of nature.

### The Lorentz Force in Action

According to the laws of electromagnetism, any charge $q$ moving with velocity $\vec{v}$ in a magnetic field $\vec{B}$ experiences a magnetic Lorentz force given by:
$$ \vec{F}_m = q(\vec{v} 	imes \vec{B}) $$

Let's apply the right-hand rule to find the direction of this force. Point your fingers in the direction of velocity (right), and curl them in the direction of the magnetic field (outwards). Your thumb points downwards. This is the direction of the force on a **positive** charge.

Since the free carriers in a metal are electrons (which are negatively charged), they experience a force in the opposite direction—upwards. As a result, the free electrons are swept towards the upper ends of the loop (towards $A$ and $B$). This migration leaves a deficit of electrons at the bottom, giving the lower ends ($C$ and $D$) a net positive charge.

### The Birth of the Electric Field

Will these electrons keep piling up at the top forever? Not at all. Nature loves equilibrium.

As the charges separate—positive at the bottom, negative at the top—they create their own internal electrostatic field, $\vec{E}$. By convention, electric field lines point from positive to negative, so this induced electric field points straight up.

This newly born electric field exerts an electric force $\vec{F}_e = q\vec{E}$ on the charges, pushing them in the exact opposite direction of the magnetic force. Very quickly, a steady state is reached where the electric force perfectly balances the magnetic force:
$$ \vec{F}_e + \vec{F}_m = 0 $$
$$ q\vec{E} + q(\vec{v} 	imes \vec{B}) = 0 $$
$$ \vec{E} = -(\vec{v} 	imes \vec{B}) $$

### The Final Verdict

This induced electric field $\vec{E}$ is not confined to just one part of the loop. It exists everywhere within the conductor that is moving through the magnetic field.

Since both the vertical arms $AD$ and $BC$ are moving with the same velocity $\vec{v}$ through the same magnetic field $\vec{B}$, the exact same charge separation and subsequent electric field induction occur in both of them.

Therefore, the electric field is induced in **both $AD$ and $BC$**. This beautiful balance of forces is what ultimately gives rise to motional EMF!

Animated Solution for Physics - Electromagnetic Induction: A metallic square loop $A B C D$ is moving in its own plane with velocity $v$ in a uniform magnetic field perpendicular to its plane as shown in the figure. Electric field is induced

Select Answer:

Visualized Solution

$The Moving Loop$

$Magnetic Force on Charges$

$Charge Accumulation$

$Induced Electric Field$

$Conclusion$

The Sigma Insight: Motional EMF

The Moving Conductor

The Lorentz Force in Action

The Birth of the Electric Field

The Final Verdict

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