KCET · Physics · Magnetic Effects of Current
In the loop shown, the magnetic induction at the point \(\mathrm{O}\) is
- A zero
- B \(\frac{\mu_{0} I}{8}\left(\frac{R_{1}-R_{2}}{R_{1} R_{2}}\right)\)
- C \(\frac{\mu_{0} \mathrm{I}}{8}\left(\frac{\mathrm{R}_{1}+\mathrm{R}_{2}}{\mathrm{R}_{\mathrm{1}} \mathrm{R}_{2}}\right)\)
- D \(\frac{\mu_{0} \mathrm{I}}{8}\left(\frac{\mathrm{R}_{1} \mathrm{R}_{2}}{\mathrm{R}_{1}+\mathrm{R}_{2}}\right)\)
Answer & Solution
Correct Answer
(C) \(\frac{\mu_{0} \mathrm{I}}{8}\left(\frac{\mathrm{R}_{1}+\mathrm{R}_{2}}{\mathrm{R}_{\mathrm{1}} \mathrm{R}_{2}}\right)\)
Step-by-step Solution
Detailed explanation
Net magnetic field at \(O\)
\(\begin{aligned}B &=B_{1}+B_{2}=\frac{-\pi / 2}{2 \pi}-\frac{\mu_{0} I}{2 R_{1}}+\frac{\pi / 2}{2 \pi} \cdot \frac{\mu_{0} I}{2 R_{2}} \\&=\frac{\mu_{0} I}{8}\left(\frac{R_{1}+R_{2}}{R_{1} R_{2}}\right)\end{aligned}\)
\(\begin{aligned}B &=B_{1}+B_{2}=\frac{-\pi / 2}{2 \pi}-\frac{\mu_{0} I}{2 R_{1}}+\frac{\pi / 2}{2 \pi} \cdot \frac{\mu_{0} I}{2 R_{2}} \\&=\frac{\mu_{0} I}{8}\left(\frac{R_{1}+R_{2}}{R_{1} R_{2}}\right)\end{aligned}\)
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