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 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)
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The field at the centre of an arc is mu0 i over 4 pi r times the angle.
Updated On: Oct 1, 2026
- \(\frac{μ_0i}{4r}\)
- \(\frac{μ_0i}{8r}\)
- \(\frac{μ_0i}{16r}\)
- \(\frac{μ_0i}{20r}\)
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The Correct Option is D
Solution and Explanation
Step 1: Understanding the Concept
A full circle of radius \(r\) gives \(B=\dfrac{\mu_0i}{2r}\) at the centre. For an arc, the field is in proportion to the angle subtended.
Step 2: Formula for an arc
\[ B=\frac{\mu_0i}{4\pi r}\,\theta \]
Step 3: Substitute \(\theta=\dfrac\pi5\)
\[ B=\frac{\mu_0i}{4\pi r}\cdot\frac\pi5=\frac{\mu_0i}{20r} \]
Step 4: Check with the fraction of a circle
The angle \(\frac\pi5\) is \(\frac1{10}\) of \(2\pi\). So the field is \(\frac1{10}\) of \(\frac{\mu_0i}{2r}\), which is \(\frac{\mu_0i}{20r}\). Both ways give 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). \[ \boxed{\frac{\mu_0i}{20r}} \]
A full circle of radius \(r\) gives \(B=\dfrac{\mu_0i}{2r}\) at the centre. For an arc, the field is in proportion to the angle subtended.
Step 2: Formula for an arc
\[ B=\frac{\mu_0i}{4\pi r}\,\theta \]
Step 3: Substitute \(\theta=\dfrac\pi5\)
\[ B=\frac{\mu_0i}{4\pi r}\cdot\frac\pi5=\frac{\mu_0i}{20r} \]
Step 4: Check with the fraction of a circle
The angle \(\frac\pi5\) is \(\frac1{10}\) of \(2\pi\). So the field is \(\frac1{10}\) of \(\frac{\mu_0i}{2r}\), which is \(\frac{\mu_0i}{20r}\). Both ways give 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). \[ \boxed{\frac{\mu_0i}{20r}} \]
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