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Question
Use (i) the Ampere’s law for H and (ii) continuity of lines of B, to conclude that inside a bar magnet, (a) lines of H run from the N pole to S pole, while (b) lines of B must run from the S pole to N pole.
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Solution
Let us consider a magnetic field line of B through the bar magnet as given in the figure below. It must be a closed loop.
Consider an Amperian loop C inside and outside the amgnet NS on side PQ of magnet then `int_P^Q vecH .vec(dl) = int_Q^P vecB/mu_0 vec(dl)`
Where B is magnetic field and m0 is dipole moment. As angle between B and dl varies from 90°, 0, 90° from R to T in figure, so cos θ is greater than 1. So `int_P^Q vecH. vec(dl) = int_Q^P vecB/mu_0 .vec(dl) > 0` i.e. posotive.
Hence, the value of B must be varied from south pole to north pole inside the magnet.
According to Ampere's law `oint_(PQP) vecH.vec(dl) = 0`
`oint_(pQP) vecH.vec(dl) = int_P^Q vecH.vec(dl) + int_Q^P vecH.vec(dl) = 0`
As `int_P^Q H.dl > 0` (outside the magnet) and `int_Q^P H.dl > 0` (inside the magnet). It is due to the angle between H and dl is more than 90° inside the magnet so cos θ is negative. It means the lines of H must run from north pole to south pole.
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