First slide

Magnetic field due to a current carrying circular ring

Question

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 $θ$ at the centre. The value of the magnetic induction at the centre due to the current in the ring is

Moderate

A

$Proportional to 2 (180^{\circ} - θ)$
B

Inversely proportional to r
C

$Zero, only if θ = 180^{\circ}$
D

Zero for all values of $θ$

Solution

Directions of currents in two parts are different, so directions of magnetic fields due to these currents are opposite. Also, applying Ohm's law across AB,
$i_{1} R_{1} = i_{2} R_{2} \Rightarrow i_{1} l_{2} = i_{2} l_{2}$
$\begin{matrix} (∵ R = ρ \frac{l}{A}) \\ Also, B_{1} = \frac{μ_{0}}{4 π} \times \frac{i_{1} l_{1}}{r^{2}} and B_{2} = \frac{μ_{0}}{4 π} \times \frac{i_{2} l_{2}}{r^{2}} (∵ l = rθ) \\ ∴ \frac{B_{2}}{B_{1}} = \frac{i_{1} l_{1}}{i_{2} l_{2}} = 1 \end{matrix}$
Hence, two field induction's are equal but of opposite direction. So, resultant magnetic induction at the centre is zero and is independent of $θ$ .

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