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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 0≤θ≤π2 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=μ0I4π∫π20dθb+cθπ ⊙ B=μ0I4ππc[ln(b+cθπ)]π20=μ0I4c[ln(b+c2)−ln(b)] B=μ0I4c[ln(1+c2b)] ⊙

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