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A loop, carring a current i, lying in th...

A loop, carring a current i, lying in the plane of the paper, is in the field of a long straight wire with current `i_0` (inward) as shown in Fig. Find the torque acting on the loop.

Text Solution

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The correct Answer is:
`(mu_0i_0i)/pi [sin theta](b-a)`
Here the field is tangential at every point on the circular portions
and hence the forces acting on these segments are zero.
Now consider two small elements of length dr at a distance r
from the current element symmetrically as shown in figure.

The magnitude of the force experienced by each element is
`dF=iBdr ( :' dvecF=ivec(dl)xxvecB)`
`{where B=(mu_0i_0)/(2pir)}`
`dF=(mu_0ii_0)/(2pir)dr`
Force on fg will be in inward direction and on eh will be in out-
ward direction.
So, torque about x-axis: `dtau=(2r sin theta)(mu_0i_0i)/(2pir)dr`
To obtain the total torque, we integrate the above expression
within proper limit
`tau=(mu_0i_0i)/pi sin theta int_a^bdr=(mu_0i_0i)/pi[sin theta](b-a)`
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