Magnetic Effects Of Current Question 51
Question: 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]
Options:
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 $
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Answer:
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} $ $ ( \because \ R=\rho \frac{l}{A} ) $ 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 $ .
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