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A battery is connected between two point...

A battery is connected between two points `A and B` on the circumference of a uniform conducting ring of radius `r` and resistance `R` . One of the arcs `AB` of the ring subtends an angle `theta` at the centre . The value of the magnetic induction at the centre due to the current in the ring is

A

proportional to `2 ( 180@ - theta)`

B

inversely proportional to `r`

C

`zero` , only if `theta = 180@`

D

`zero` for all values of `theta`

Text Solution

Verified by Experts

The correct Answer is:
D
Magnetic field at the centre due to current in arc ` ABC` is
`b_(1) = (mu_(0))/(4 pi) (I_(1)/( r) theta` ( directed upwards)
Magnetic field at the centre due to current in arc ` ADB` is
` B_(2) = (mu)/(4 pi) (I_(2))/(r) ( 2 pi - theta )` ( Directed downwards)
Therefore net magnetic field at the centre
` B = (mu)/(4 pi) (I_(1))/(r) ( theta)/( pi) - (mu)/(4 pi) ( I_(2))/(r) ( 2 pi - theta )` ltbr gt Also , I_(1_ = ( E)/(R_(1)) = (E) /( rhol_(1))//A = ( EA)/( rho r theta )`
and ` I_(2) = (E)/(R_(2)) = (rho l _(2) )//(A) = (EA)/( rho r( 2 pi - theta))`
:. ` = (mu)/(4 pi) [ (EA)/( rho r theta) xx (theta)/(r) - ( EA)/( rho r ( 2 pi - theta )) xx ( 2 pi - theta )/( r ) ] = 0 `

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