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\[\theta \].You need to login to perform this action.
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