A current loop lies in the xy-plane; its boundary consists of a square of side a and four outward semicircular arcs of diameter a as illustrated. With k̂ out of the page, its magnetic moment is (π/n + 1)a²I k̂. Find n.
Physics · Mixed Chapter
JEE Mains

Question

A current loop lies in the xy-plane; its boundary consists of a square of side a and four outward semicircular arcs of diameter a as illustrated. With k̂ out of the page, its magnetic moment is (π/n + 1)a²I k̂. Find n.

Physics · Mixed Chapter

G1 Physics · Accepted Answer

Step 1 — Magnetic dipole moment of a closed planar current loop: μ = IA, directed perpendicular to the loop according to the right-hand rule. Step 2 — Area of the loop: The shaded/enclosed area consists of a square of side a plus four outward semicircles, each of radius a/2. A = a² + 4[½π(a/2)²] = a² + (π/2)a² = (1 + π/2)a². Step 3 — Magnetic moment: 𝐌 = I(1 + π/2)a²k̂. Step 4 — Compare with (1 + π/n)a²Ik̂: π/n = π/2 ⇒ n = 2. Answer: 2.