A conducting square loop of side L, mass M and resistance R enters a region y ≥ 0 containing uniform magnetic field B = B₀ k̂ with initial velocity v₀ ĵ. Define K = B₀²L²/(RM). Ignoring self-inductance and gravity, which statements are correct?
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Question

A conducting square loop of side L, mass M and resistance R enters a region y ≥ 0 containing uniform magnetic field B = B₀ k̂ with initial velocity v₀ ĵ. Define K = B₀²L²/(RM). Ignoring self-inductance and gravity, which statements are correct?

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G1 Physics · Accepted Answer

While the loop is partially entering, magnetic drag is

F = −(B₀²L²/R)v = −MKv.

Thus dv/dt = −Kv and, using dv/dt = v dv/dx, dv/dx = −K ⇒ v = v₀ − Kx.

The loop fully enters when x = L.

A: for v₀ = 1.5KL, v(L) = 0.5KL > 0, so it does not stop before entering fully. False.

B: once the loop is completely inside a uniform field, the flux no longer changes; induced current and magnetic drag become zero. True.

C: for v₀ = KL/10, stopping distance v₀/K = L/10. The speed approaches zero asymptotically; the stated finite stopping time is not correct. False.

D: x(t) = (v₀/K)(1 − e⁻ᴷᵗ). With v₀ = 3KL and x = L, 1 = 3(1 − e⁻ᴷᵗ) ⇒ e⁻ᴷᵗ = 2/3 ⇒ t = (1/K)ln(3/2). True.

Correct options: B, D.