Question

Derive an expression for magnetic field at a point on the axial line of a bar magnet.

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Solution
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Let be a bar magnet of magnetic length and having each pole of magnetic strength . O is the center of magnet and is a point on axial line at a distance from the center of magnet , at which magnetic field has to be measured .
The magnetic field at due to pole of magnet ,
     
or     (along PX) .......................eq1
And , the magnetic field at due to pole of magnet ,
     
or     (along PS) ......................eq2

Therefore , resultant magnetic field at point ,
         (-ive sign is due to opposite directionS of and )
It is clear from eq1 and eq2 that ,therefore the direction of   will be along .
or         (alongPX)
or         (alongPX)
Now , (magnetic dipole moment of magnet)
Hence ,     (alongPX)

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