question_answer

A battery is connected between two points A and B on the circumference of a uniform conducting ring of radius r and resistance R. One of the arcs AB of the ring subtends an angle \[\theta \] at the centre. The value of the magnetic induction at the centre due to the current in the ring is [IIT 1995]

A)            Proportional to \[2\,(180{}^\circ -\theta )\]

B)            Inversely proportional to r

C)            Zero, only if \[\theta =180{}^\circ \]

D)            Zero for all values of \[\theta \]

Correct Answer: D

Solution :

                   Directions of currents in two parts are different, so directions of magnetic fields due to these currents are opposite. Also applying Ohm?s law across \[AB\] \[{{i}_{1}}{{R}_{1}}={{i}_{2}}{{R}_{2}}\Rightarrow {{i}_{1}}{{l}_{2}}={{i}_{2}}{{l}_{2}}\] \[\left( \because \ R=\rho \frac{l}{A} \right)\] Also \[{{B}_{1}}=\frac{{{\mu }_{o}}}{4\pi }\times \frac{{{i}_{1}}{{l}_{1}}}{{{r}^{2}}}\] and \[{{B}_{2}}=\frac{{{\mu }_{o}}}{4\pi }\times \frac{{{i}_{2}}{{l}_{2}}}{{{r}^{2}}}\] (\[\because \ l=r\theta \])  \[\therefore \,\frac{{{B}_{2}}}{{{B}_{1}}}=\frac{{{i}_{1}}{{l}_{1}}}{{{i}_{2}}{{l}_{2}}}=1\] Hence, two field induction?s are equal but of opposite direction. So, resultant magnetic induction at the centre is zero and is independent of\[\theta \].