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Question: Two identical magnetic dipoles of magnetic moment 2A m2 are placed at a separation of 2 m with their...

Two identical magnetic dipoles of magnetic moment 2A m2 are placed at a separation of 2 m with their axes perpendicular to each other in air. The resultant magnetic field at a midpoint between the dipoles is

A

45×10−5 T4 \sqrt { 5 } \times 10 ^ { - 5 } \mathrm {~T}

B

25×10−5 T2 \sqrt { 5 } \times 10 ^ { - 5 } \mathrm {~T}

C

45×10−7 T4 \sqrt { 5 } \times 10 ^ { - 7 } \mathrm {~T}

D

25×10−7 T2 \sqrt { 5 } \times 10 ^ { - 7 } \mathrm {~T}

Answer

25×10−7 T2 \sqrt { 5 } \times 10 ^ { - 7 } \mathrm {~T}

Explanation

Solution

:

Let point P be a midpoint between the dipoles. The point P will be in end – on position with respect to one dipole and in broad – side on position with respect to the other.

∴B1=μ04π2 m1r13=10−7×2×2(1)3=4×10−7 T\therefore \mathrm { B } _ { 1 } = \frac { \mu _ { 0 } } { 4 \pi } \frac { 2 \mathrm {~m} _ { 1 } } { \mathrm { r } _ { 1 } ^ { 3 } } = \frac { 10 ^ { - 7 } \times 2 \times 2 } { ( 1 ) ^ { 3 } } = 4 \times 10 ^ { - 7 } \mathrm {~T}

and, B2=μ04πm2r23=10−7×2(1)3=2×10−7 T\mathrm { B } _ { 2 } = \frac { \mu _ { 0 } } { 4 \pi } \frac { \mathrm { m } _ { 2 } } { \mathrm { r } _ { 2 } ^ { 3 } } = \frac { 10 ^ { - 7 } \times 2 } { ( 1 ) ^ { 3 } } = 2 \times 10 ^ { - 7 } \mathrm {~T}

As and are perpendicular to each other, therefore the resultant magnetic field at point P is.

=10−716+4=10−720=25×10−7 T= 10 ^ { - 7 } \sqrt { 16 + 4 } = 10 ^ { - 7 } \sqrt { 20 } = 2 \sqrt { 5 } \times 10 ^ { - 7 } \mathrm {~T}