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Question

The magnetic field at the origin due to the current flowing in the wire is 



Your Answer
A
μ0I8πa(^i+^k)
Your Answer
B
μ0I2πa(^i+^k)
Correct Answer
C
μ0I8πa(^i+^k)
Your Answer
D
μ0I4πa2(^i^k)

Solution

The correct option is A μ0I8πa(^i+^k)
One part of the wire is lying along positive x-axis, one part along the line x=z and one part along positive y-axis.


There is no magnetic field at the origin due to the segments 1 and 2 (since they both pass through the origin). Hence, net magnetic field at origin is only due to 3.

The magnetic field due to 3 at origin is given by the formula for the magnetic field due to a semi infinite wire at a point.



B=μ0i4πd(sinα+sinβ) where α and β are the angles made by the line joining the point O with the extremes of the wire.
Here, α=0,β=90
B=μ0i4πd

To find the direction of field, plot the magnetic field lines around the wire passing through point O.


Then, we can say that the magnetic field at O must be perpendicular to the radius of the circle passing through O shown in the diagram.
Since the radius is along the line x=z, the line perpendicular to it is x=z i.e the direction of magnetic field at O is along (^i+^k)
Thus, magnetic field at O can be written as
B=μ0i4πdcos45(^i)+μ0i4πdsin45(^k)
=μ0i4π2acos45(^i+^k)
=μ0i8πa(^i+^k)

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