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Question

# A wire carrying current i is in the shape of a plane curve given by r=f(θ) where the r and θ are polar coordinates. The magnetic field at the center of curvature of the current carrying wire is

A
μ0i4πsinθdθr
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B
μ0i4πsinθdθr2
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C
μ0i2πsinθdθr
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D
μ0i2πsinθdθr2
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Solution

## The correct option is A μ0i4π∫sinθdθrHere, C is the center of curvature of the portion of wire. Using biot. Savart's law: −→dB=μ0i4π→dl×→rr3 ⇒dB=μ0i(dl×r×sinθ)4πr3 or, dB=μ0idlsinθ4πr2...(i) From the figure shown, dl=rdθ....(ii) From eq. (i) and (ii) dB=μ0irdθsinθ4πr2=μ0i4πrsinθdθ ⇒B=∫dB=μ0i4π∫sinθdθr

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