Magnetic field

A square wire loop having side length \(a\), mass \(m\), and resistance \(R\) is moving along the positive \(x\)-axis at a speed of \(v_0\). It enters a uniform, steady magnetic field \(\vec B = B_0 \big(-\widehat k \big) \) at \(t = 0 \) seconds, as shown in the figure.

Find the total amount of heat loss in the resistance.

Details and Assumptions:

• The magnitude of \(v_0 \) is sufficient that the loop comes out of the region of magnetic field with some speed.

• Neglect any type of energy loss other than the heat loss in resistance of the wires of the square.

• Take \( v_0 = \dfrac{3B^2 a^3}{mR} \).