Question

A region in the form of an equilateral triangle (in x-y plane) of height L has a uniform magnetic field 𝐵⃗ pointing in the +z-direction. A conducting loop PQR, in the form of an equilateral triangle of the same height 𝐿, is placed in the x-y plane with its vertex P at x = 0 in the orientation shown in the figure. At 𝑡 = 0, the loop starts entering the region of the magnetic field with a uniform velocity 𝑣 along the +x-direction. The plane of the loop and its orientation remain unchanged throughout its motion.

Which of the following graph best depicts the variation of the induced emf (E) in the loop as a function of the distance (𝑥) starting from 𝑥 = 0?

• A.
• B.
• C.
• D.

Answer 1

The Correct Option is A

Flux and Electric Field Calculation 

For any time \( t \), assume \( x < L \):

The area is:

\[ \text{Area} = \frac{1}{2} \cdot x \cdot x \cdot \tan(30^\circ) \cdot 4 = \frac{1}{2} x^2 \tan(30^\circ) \]

The flux is:

\[ \Phi = B_0 \cdot \text{Area} \quad \Rightarrow \quad E = -\frac{d\Phi}{dt} = -Bv \cdot \tan(30^\circ) \cdot x \]

Thus, we have:

\[ E \propto -x \]

When \( x \geq L \),

Recompute using the difference in areas and find the new expression for \( E \), which changes direction.

Therefore, the correct graph is Option (A).