A circular loop of radius r carries a current i. How should a long, straight wire carrying a current, 4i be placed in the plane of the circle so that the magnetic field at the centre becomes zero ?

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Solution

Magnetic field due to loop at its centre

$B = \frac{μ_{0} i}{2 r}$

To vanish above magnetic field we place a straight wire of 4i current at a distance x such that their magnetic fields directions be opposite.

Thus $\frac{μ_{0} i}{2 r} = \frac{μ_{0} 4 i}{2 π x}$

$\Rightarrow x = \frac{8 x}{2 π} = \frac{4 r}{π}$

i.e wire is placed $\frac{4 r}{π}$ from the centre of the loop

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A circular loop of radius r carries a current i. How should a long, straight wire carrying a current, 4i be placed in the plane of the circle so that the magnetic field at the centre becomes zero ?

Magnetic field due to loop at its centre B=μ0i2r To vanish above magnetic field we place a straight wire of 4i current at a distance x such that their magnetic fields directions be opposite. Thus μ0i2r=μ0 4i2πx ⇒ x=8x2π=4rπ i.e wire is placed 4rπ from the centre of the loop