currents\[{{i}_{1}}\]and\[{{i}_{2}}\]as shown in the diagram. If ratio of their radii is 1 : 2 and ratio of the flux densities at O due to A and is 1 : 3 then the value of \[\frac{{{i}_{1}}}{{{i}_{2}}}\] will be:
A) \[\frac{1}{2}\]
B) \[\frac{1}{3}\]
C) \[\frac{1}{4}\]
D) \[\frac{1}{6}\]
Correct Answer: D
Solution :
Ratio of radii of the two conductors \[{{r}_{A}}:{{r}_{B}}=1:2\] Ratio of the flux densities at the centre\[O\,{{B}_{A}}:B{{ & }_{B}}=1:3\] The magnitude of the flux density at the centre of a circular current carrying conductor is given by \[B=\frac{{{\mu }_{0}}i}{2r}\propto \frac{1}{r}\] Hence \[\frac{{{B}_{1}}}{{{B}_{2}}}=\frac{{{r}_{2}}}{{{r}_{1}}}\times \frac{{{i}_{1}}}{{{i}_{2}}}\] \[\frac{1}{3}=\frac{{{i}_{1}}}{{{i}_{2}}}\times \frac{2}{1}or\frac{{{i}_{1}}}{{{i}_{2}}}=\frac{1}{6}\]
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