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Three semi-infinite mutually perpendicul...

Three semi-infinite mutually perpendicular conductors are joined at the origin O as shown in figure. A current 2I enters through the conductor along Z-axis towards the origin O and leaves through the other two as shown in the figure. What is the magnetic field at a point P with position vector `vec(r ) = hat(i) + hat(j) + hat(k)` ?

Text Solution

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Visualize the geometry in three dimensions. Field due to current on Z-axis
`vec(B_(z)) = (mu_(0))/(4 pi) xx 2I [sin ((pi)/(2)) - sin (sin^(-1) ((1)/(sqrt(3))))] (sin 45^(@) hat(i) - cos 45^(@) hat(j))`
`vec(B_(x)) = (mu_(0))/(4 pi) xx I [sin ((pi)/(2)) + sin (sin^(-1) ((1)/(sqrt(3))))] (cos 45^(@) hat(j) + sin 45^(@) hat(k))`
`vec(B_(y)) = (mu_(0))/(4 pi) xx I [sin ((pi)/(2)) + sin (sin^(-1) ((1)/(sqrt(3))))](cos 45^(@) hat(i) - sin 45^(@) hat(k))`
`vec(B) = vec(B_(X)) + vec(B_(z)) = (mu_(0))/(4 sqrt(2)) [sqrt(3) - 1](-hat(i) + hat(j))`
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