JEE Physics 2025 Question with Solution

Given: A rectangular metallic loop moves out of a uniform magnetic field with constant speed.

Find: The correct graph of the magnitude of induced emf $|\varepsilon|$ versus time $t$ when the loop is partially inside the magnetic field.

Using Faraday's law:

$$\varepsilon = \left| \frac{d\Phi_B}{dt} \right|$$

When the loop is partially inside the field, the magnetic flux through it changes because the area inside the field changes with time.

If the loop moves with constant speed $v$, then the length leaving the field changes uniformly with time, so the area inside the field also changes uniformly. Hence,

$$\Phi_B = B l x$$

where $B$ is the magnetic field, $l$ is the side length perpendicular to motion, and $x$ is the portion inside the field.

Differentiating with respect to time:

$$\varepsilon = B l \frac{dx}{dt} = B l v$$

Since $B$, $l$, and $v$ are constant, the magnitude of induced emf remains constant while the loop is partially inside the field.

Therefore, the correct graph is a horizontal line of constant magnitude. The correct option is D (Option $4$).

When the loop is completely inside or completely outside the magnetic field region, the flux does not change, so induced emf is zero.

During the interval in which the loop is leaving the field, the overlap area decreases linearly with time because the speed is constant. Therefore flux decreases linearly with time.

If flux varies linearly with time, then its time derivative has constant magnitude. Hence,

$$|\varepsilon| = \left| \frac{d\Phi_B}{dt} \right| = \text{constant}$$

So the required graph during partial overlap is the constant-emf graph, corresponding to Option $4$.