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Inside a long straight uniform wire of r...

Inside a long straight uniform wire of round cross-section, there is a long round cylindrical cavity whose axis is parallel to the axis of the wire and displaced from the latter by a distance l. A direct current of density j flows along the wire. Find the magnetic induction inside the cavity. Consider, in particular, the case `l=0`.

A

0

B

`1/2mu_(0) vecJ xx vecd`

C

`mu_(0) vecJ xx vecd`

D

`3/2mu_(0) vecJ xx vecd`

Text Solution

Verified by Experts

The correct Answer is:
B
From Ampere's law, `oint vecB.dvecl = mu_(0)int vecJ. dvecs`
Inside the conductor at a distance r from itx axis.
`B(2pir) = mu_(0)J(pir^(2)), B=1/2 mu_(0)Jr`
Vectorially, `vecB = 1/2 mu_(0)vecJ xx vecr`
Now, conductor has a cavity so we can assume that current in it is the superposition of positive current and negative current (in place of cavity).
Required magnetic field at point M
`vecB_(N)` = Magnetic field at M due to whole conductor- Magnetic field at M due to cavity shaped conductor
`vecB_(N) = 1/2 mu_(0)(vecJ xx vec(OM)) - 1/2 mu_(0)(vecj xx vec(PM))`
`=12 mu_(0)vecj xx (vec(OM) - vec(PM))=1/2 mu_(0)vecj xx vecd`.
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