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 ${\theta }^o$ at the centre. The value of the magnetic induction at the centre due to the current in the ring is

• A. proportional to $2({180}^o-{\theta }^o)$
• B. inversely proportional to $r$
• C. zero, only if $\theta ={180}^o$
  
• D. zero for all values of $\theta$

Answer 1

The correct option is D zero for all values of $\theta$

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 $$AB$$ $$i_1{R}_1=i_2{R}_2\Rightarrow i_1{l}_2=i_2{l}_2$$ $$(\because R=\rho \frac{l}{A})$$ Also $$B_1=\frac{{\mu }_o}{4\pi }\times \frac{i_1{l}_1}{r^2}$$ and $$B_2=\frac{{\mu }_o}{4\pi }\times \frac{i_2{l}_2}{r^2}$$ ($$\because l=r\theta$$) $$\therefore \frac{B_2}{B_1}=\frac{i_1{l}_1}{i_2{l}_2}=1$$
Hence, two fields are equal but in opposite direction. So, resultant magnetic induction at the centre is zero and is independent of$$\theta$$.