First slide

Magnetic field due to current element

Question

A square conducting loop of length ‘L’ carries a current ‘i’. the magnetic field at the center of the loop is

Moderate

A

Independent of L
B

Proportional to L
C

Inversely proportional to L
D

directly propotional to square of L

Solution

Magnetic field at the center due to either arm
$B_{1} = \frac{μ_{0}}{4 π} \times \frac{i}{(L / 2)} [\sin 45^{0} + \sin 45^{0}]$ $= \frac{μ_{0}}{4 π} \times \frac{2 \sqrt{2} i}{L}$
Field at center due to the four arms of the square
$B = 4 B_{1} = 4 \times \frac{μ_{0}}{4 π} \times \frac{2 \sqrt{2} i}{L}$
$\Rightarrow B ∝ \frac{1}{L}$ .

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