Q. A cell is connected between two points of a uniformly thick circular conductor. The magnetic field at the centre of the lop will be (Here $i_{1}$ and $i_{2}$ are the currents flowing in the two parts of the circular conductor of radius $'a'$ and $\mu_{0}$ has the usual meaning)
Explanation
(a) Direction of currents in two parts is different, so direction of magnetic fields due to these currents is opposite. Also applying Ohm’s law across $AB$
$i_{1}R_{1} = i_{2}R_{2} \Rightarrow i_{1}I_{2} = i_{2}I_{2}$
$\left\lbrack \because R = \rho\frac{l}{A} \right\rbrack$
Also $B_{1} = \frac{\mu_{o}}{4\pi} \times \frac{i_{1}l_{1}}{r^{2}}$ and $B_{2} = \frac{\mu_{o}}{4\pi} \times \frac{i_{2}l_{2}}{r^{2}}\ \ \lbrack\because l = r\theta\rbrack$
$\therefore\frac{B_{2}}{B_{1}} = \frac{i_{1}l_{1}}{i_{2}l_{2}} = 1$
Hence, two field induction’s are equal but of opposite direction. So, resultant magnetic induction at the centre is zero and is independent of $\theta$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.