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Question

A long solid cylindrical current carring conductor is placed along the z aixs, carrying current I in the negative z direction. The magnetic field B at a point having coordinate (x,y) on the z=0 plane is

A
μ0I(y^ix^j)2π(x2+y2)
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B
μ0I(x^i+y^j)2π(x2+y2)
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C
μ0I(x^jy^j)2π(x2+y2)
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D
μ0I(x^iy^j)2π(x2+y2)
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Solution

The correct option is A μ0I(y^ix^j)2π(x2+y2)
The plane representing z=0 is the xy plane. Let us represent the current carrying conductor in xy plane, carrying a current in z direction as shown in the figure.


The direction of magnetic field at point P will be tangential to the circular field line in clockwise direction with radius r.

Here,

B=Bsinθ^iBcosθ^j ......(i)

The magnitude of magnetic field due to solid cylinder at an outside point is,

B=μ0I2πr

Also, sinθ=yr and cosθ=xr

Substituting these values in Eq. (i)

B=μ0I2πr[sinθ^icosθ^j]

B=μ0I2πr[yr^ixr^j]

or, B=μ0I2πr2[y^ix^j]

From geometry of figure,

r2=x2+y2

Thus,
B=μ0I[y^ix^j]2π(x2+y2)

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