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Motional EMF in an Open Conductor

A conducting ...

Question

A conducting wire carrying current is arranged as shown. The magnetic field at ' $O$ '

\frac{μ_{0} i}{12} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]

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\frac{μ_{0} i}{12} [\frac{1}{R_{1}} + \frac{1}{R_{2}}]

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\frac{μ_{0} i}{6} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]

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\frac{μ_{0} i}{6} [\frac{1}{R_{1}} + \frac{1}{R_{2}}]

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Solution

The correct option is B $\frac{μ_{0} i}{12} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]$
Magnetic induction at point O, $B = \frac{μ_{o}}{4 π} . \frac{i}{R} (α)$ where $α = 60^{o} = π / 3$
Magnetic induction due to parts 3 and 4 is zero.
Magnetic induction due to part 1, $B_{1} = \frac{μ_{o}}{4 π} . \frac{i}{R_{1}} \frac{π}{3} = \frac{μ_{o} i}{12 R_{1}}$ (downwards)
Magnetic induction due to part 2, $B_{2} = \frac{μ_{o}}{4 π} . \frac{i}{R_{2}} \frac{π}{3} = \frac{μ_{o} i}{12 R_{2}}$ (upwards)

$∴$ Magnetic induction at point O, $B_{o} = B_{1} - B_{2} = \frac{μ_{o} i}{12} [\frac{1}{R_{1}} - \frac{1}{R_{2}}]$ (downwards)

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A conducting wire carrying current is arranged as shown. The magnetic field at 'O'

A conducting wire carrying current is arranged as shown. The magnetic field at ' $O$ '