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Question

A wire carrying current I is laid in shape of a curve which is represented in plane polar coordinate system as,
r=b+cπθ for 0θπ2
Here b and c are positive constants. θ is the angle measured with respect to positive X direction in anticlockwise sense and r is the distance from origin (see figure). The magnetic field at the origin due to the wire is found to be μoIxcln(1+cyb)
The value of xy is

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Solution

Unit vectors along r and perpendicular to r are ^r and ^θ respectively. A small element of length along the curve can be represented as,
dl=(dr)^r+(rdθ)^θ
Field due to such an element at O is,
dB=μ04π I[dl×(^r)]r2
dB=μ0 I4π r2[((dr)^r+(rdθ)^θ)×(^r)]
dB=μ0I4π rdθ ^k

Magnetic field B
B=μ0I4ππ20dθb+cθπ
B=μ0I4ππc[ln(b+cθπ)]π20=μ0I4c[ln(b+c2)ln(b)]

B=μ0I4c[ln(1+c2b)]

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