A conducting wire of parabolic shape, initially y = x², is moving with velocity $$v = {v_0}\widehat i$$ in a non-uniform magnetic field $$B = {B_0}\left( {1 + {{\left( {{y \over L}} \right)}^\beta }} \right)\widehat k$$, as shown in figure. If V₀, B₀, L and $$\beta $$ are positive constants and $$\Delta $$$$\phi $$ is the potential difference developed between the ends of the wire, then the correct statement(s) is/are

In the figure below, the switches $${S_1}$$ and $${S_2}$$ are closed simultaneously at $$t=0$$ and a current starts to flow in the circuit. Both the batteries have the same magnitude of the electromotive force (emf) and the polarities are as indicated in the figure. Ignore mutual inductance between the inductors. The current $$I$$ in the middle wire reaches its maximum magnitude $${I_{\max }}$$ at time $$t = \tau $$ . Which of the following statements is (are) true?

A circular insulated copper wire loop is twisted to form two loops of area $$A$$ and $$2A$$ as shown in the figure. At the point of crossing the wires remain electrically insulated from each other. The entire loop lies in the plane (of the paper). A uniform magnetic field $$\overrightarrow B $$ points into the plane of the paper. At $$t=0,$$ the loop starts rotating about the common diameter as axis with a constant angular velocity $$\omega $$ in the magnetic field.

Which of the following options is/are correct?

A rigid wire loop of square shape having side of length L and resistance R is moving along the X-axis with a constant velocity v₀ in the plane of the paper. At t = 0, the right edge of the loop enters a region of length 3L where there is a uniform magnetic field B₀ into the plane of the paper, as shown in the figure. For sufficiently large v₀, the loop eventually crosses the region. Let x be the location of the right edge of the loop. Let v(x), I(x) and F(x) represent the velocity of the loop, current in the loop, and force on the loop, respectively, as a function of x. Counter-clockwise current is taken as positive.


Which of the following schematic plot(s) is (are) correct? (Ignore gravity)