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 $A B$ 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 $2 \left( 180 ^ { \circ } - \theta \right)$
• B. Inversely proportional to r
• C. Zero, only if $\theta = 180 ^ { \circ }$
• D. Zero for all values of $\theta$

Answer 1

Answer: (D)

Zero for all values of $\theta$

Answer 2

Directions of currents in two parts are different, so directions of magnetic fields due to these currents are different.

Also applying Ohm's law across $A B$

$i _ { 1 } R _ { 1 } = i _ { 2 } R _ { 2 } \Rightarrow i _ { 1 } l _ { 1 } = i _ { 2 } l _ { 2 }$ …..(i)

Also $B _ { 1 } = \frac { \mu _ { 0 } } { 4 \pi } \times \frac { i _ { 1 } l _ { 1 } } { r ^ { 2 } }$ and ;

∴ $\frac { B _ { 2 } } { B _ { 1 } } = \frac { i _ { 1 } l _ { 1 } } { i _ { 2 } l _ { 2 } } = 1$ [Using (i)]

Hence, two field are equal but of opposite direction. So, resultant magnetic induction at the centre is zero and is independent of $\theta$ .