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A battery is connected between two points A and B on the circumference of a uniform conducting ring of radius r andresistance 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 battery is connected between two points A and B on the circumference of a uniform conducting ring of radius r andresistance 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
Updated On: Jun 14, 2022
- proportional to $(180^\circ-\theta)$
- inversely proportional to r
- zero, only if $(\theta=180^\circ)$
- zero for all values of $\theta$
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The Correct Option is D
Solution and Explanation
For a current flowing into a circular arc, the magnetic induction at the centre is
$B=\Bigg(\frac{\mu_0 i}{4\pi r}\Bigg)\theta$
$or \, \, \, \, \, \, \, \, \, B ? i \theta$
In the given problem, the total current is divided into two arcs
$\hspace15mm i ? \frac{1}{resistance of arc} ?\frac{1}{length of arc }$
$\hspace15mm ?\frac{1}{angle subtended at centre(\theta)}$
$or \, \, \, \, \, \, \, \, \, \, i\theta= constant$
i.e. magnetic field at centre due to arc AB is equal and opposite to the magnetic field at centre due to arc ACB. Or the net magnetic field at centre is zero.
$B=\Bigg(\frac{\mu_0 i}{4\pi r}\Bigg)\theta$
$or \, \, \, \, \, \, \, \, \, B ? i \theta$
In the given problem, the total current is divided into two arcs
$\hspace15mm i ? \frac{1}{resistance of arc} ?\frac{1}{length of arc }$
$\hspace15mm ?\frac{1}{angle subtended at centre(\theta)}$
$or \, \, \, \, \, \, \, \, \, \, i\theta= constant$
i.e. magnetic field at centre due to arc AB is equal and opposite to the magnetic field at centre due to arc ACB. Or the net magnetic field at centre is zero.
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