The wire loop PQRS formed by joining two semicircular wires of radii $R_{1}$ and $R_{2}$

Carries a current I as shown. find the magnitude of the magnetic induction at the centre C?

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\frac{μ_{0} I}{2} (\frac{R_{2} - R_{1}}{R_{2} R_{1}})

\frac{μ_{0} I}{4} (\frac{R_{2} - R_{1}}{R_{2} R_{1}})

\frac{μ_{0} I}{2} (\frac{1}{R_{2}} - \frac{1}{R_{1}})

None of these

answer is B.

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Detailed Solution

The magnetic field at the centre of the semicircle upwards is given by

\frac{μ_{0} I}{4} (\frac{R_{2} - R_{1}}{R_{2} R_{1}})

Magnetic field due to semi circular arc o radius r at the center of the circle =

\frac{μ_{0} I}{2 r}

Hence magnetic field upward due to inner semicircle =

\frac{μ_{0} I}{4 R_{1}}

Magnetic field upwards due to outer semicircle =

\frac{μ_{0} I}{4 R_{2}}

Hence net magnetic field at the centre of the semicircle upwards =

\frac{μ_{0} I}{4} (\frac{R_{2} - R_{1}}{R_{2} R_{1}})

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