Question

A region in the form of an equilateral triangle (in x-y plane) of height \( L \) has a uniform magnetic field \( \mathbf{B} \) 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 \( \mathbf{v} \) along the \( +x \)-direction. The plane of the loop and its orientation remain unchanged throughout its motion. 

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

Hint:

Use Faraday's law and the concept of flux change to analyze the emf.

Answer 1

For any time \( t \), assume \( x<L \): \[ \text{Area} = \frac{1}{2} \cdot x \cdot \frac{x}{2} \cdot \tan(30^\circ) \cdot 4 = \frac{1}{2} x^2 \tan(30^\circ). \] The flux is: \[ \Phi = B_0 \cdot \text{Area} \quad \Rightarrow \quad \mathcal{E} = -\frac{d\Phi}{dt} = -Bv \cdot \tan(30^\circ) \cdot x. \] Thus, \( \mathcal{E} \propto -x \). When \( x \geq L \), recompute using the difference in areas and find the new expression for \( \mathcal{E} \), which changes direction. Therefore, the correct graph is Option (A).