Question

A circular arc of wire of radius of curvature \(r\) subtends an angle of \(\frac{π}{5}\) radian at its centre. If current \(i\) is flowing in it then the magnetic induction at its centre is (\(μ_0\) = permeability of free space)

• A. \(\frac{μ_0i}{4r}\)
• B. \(\frac{μ_0i}{8r}\)
• C. \(\frac{μ_0i}{16r}\)
• D. \(\frac{μ_0i}{20r}\)

Hint:

The field at the centre of an arc is mu0 i over 4 pi r times the angle.

Answer 1

The Correct Option is D

Step 1: Approach
Use the fraction of the full circle.

Step 2: Fraction
$\dfrac{\pi/5}{2\pi}=\dfrac1{10}$.

Step 3: Field
Full loop: $\dfrac{\mu_0i}{2r}$. Tenth of it: $\dfrac{\mu_0i}{20r}$. Option (D).

Final Answer:
The arc is one tenth of a circle, so the field is one tenth of mu0 i over 2r, which is mu0 i over 20 r, option (D). \[ \boxed{\frac{\mu_0i}{20r}} \]