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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 arc AB of the ring subtends an angle `theta` at the centre. The value of the magnetic field produced 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 `theta`

Text Solution

Verified by Experts

The correct Answer is:
D
(d) Magnetic field at the centre due to current in arc ACB is
`B_1=(mu_0)/(4pi) (2piI_1)/r theta/(2pi)` (directed upwards)
magnetic field at the centre due to current in arc ADB is
`B_2=(mu_0)/(4pi) (2piI_2)/r (2pi-theta)/(2pi)` (directed downwards)

Therefore, net magnetic field at the centre
`B=(mu_0)/(4pi) (2piI_1)/r theta/(2pi)-(mu_0)/(4pi) (2piI_2)/r (2pi-theta)/(2pi)`
Also, `I_1=E/(R_1)=E/(rhol_1//A)=(EA)/(rhor theta)`
And `I_2=E/(R_2)=E/(rhol_2//A)=(EA)/(rhor(2pi- theta))`
`:. B=(mu_0)/(4pi)[(EA)/(rhor theta)xxtheta/r-(EA)/(rhor(2pi-theta))xx((2pi-theta))/r]=0`
(d) is the correct option.
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