A circular conducting loop has radius 0.2 m and resistance 5 Ω. A uniform 0.5 T field perpendicular to its plane, directed into the page, falls to zero in 0.1 s. Find average induced emf, current and its direction.

Solution & explanation

1. Given: r = 0.2 m; R = 5 Ω; B₁ = 0.5 T; B₂ = 0; Δt = 0.1 s; one turn.

2. Find the circle area.

   $$A=\pi \times r^2=\pi \times {0.2}^2=0.04\times \pi$$

3. The field is normal to the loop, so flux = BA.

   $${\Phi }_1=0.5\times 0.04\times \pi =0.02\times \pi$$

4. Final flux is zero. Subtract initial flux.

   $${\Delta }_{\Phi }=-1\times 0.02\times \pi +0=-0.02\times \pi$$

5. Use the magnitude of flux change per second.

   $${\epsilon }_m=\frac{0.02\times \pi }{0.1}=0.2\times \pi$$

6. Apply Ohm’s law to the loop.

   $$I=\frac{0.2\times \pi }{5}=0.04\times \pi$$

7. Evaluate π only at the end.

   $$I=0.1256637061$$

8. Into-page flux decreases. The induced field must point into the page to oppose that decrease. The right-hand grip rule gives clockwise current.

9. State the result with units: Average emf ≈ 0.628 V; current ≈ 0.126 A clockwise, viewed on the page.

ε ≈ 0.628 V; I ≈ 0.126 A clockwise