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Question

# A long straight wire along the Z-axis carries a current ′I′ in the negative Z-direction. The induced magnetic field B at a point having coordinates (x,y) is:

A
μ0I2π(x^iy^j)(x2+y2)
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B
μ0I2π(x^jy^i)(x2+y2)
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C
μ0I2π(x^i+y^j)(x2+y2)
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D
μ0I2π(y^ix^j)(x2+y2)
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Solution

## The correct option is D μ0I2π(y^i−x^j)(x2+y2)According to Biot Savart Law,→B(r)=μo4π∫CI→dl×→rr3→dl=−dz^k→r=x^i+y^j−z^k r=(x2+y2+z2)1/2Let d2=x2+y2→B=μoI4π∫∞z=−∞(−dz^k)×(x^i+y^j−z^k)(d2+z2)3/2→B=μoI4π∫∞z=−∞−x dz^j+y dz^i(d2+z2)3/2Let z=dtan(θ)Then, dz=dsec2(θ)dθ→B=μoI4π∫π/2θ=−π/2−xdcos(θ)dθ^j+ydcos(θ)dθ^id3Solving the integration gives,→B=μoI(−x^j+y^i)2π(x2+y2)

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