Question

Two concentric coplanar circular loops made of wire, with resistance per unit length $10^{-4} \Omega \mathrm{m}^{-1}$ have diameters $0.2 \mathrm{~m}$ and $2 \mathrm{~m}$. A time varying potential difference $(4+2.5 \mathrm{t})$ is applied to the larger loop. Calculate the current in the smaller loop.
(A) $0.25 \mathrm{~A}$
(B) $0.75~\text{A}$
(C) $1.25 \mathrm{~A}$
(D) 1.75 A

Answer 1

Hint: Some flux lines will be generated around the conductor as it is carrying current (DC), and they are concentric with the conductor's central axis. Calculate I and then put the value of I in the equation $B=\dfrac{\mu_{0} I}{2 R}$. Calculate Flux and the induced emf in the circular loop then take out the current in the smaller loop by the help of the bigger loop.

Complete Step-by-Step solution:
The magnetic field at the centre $\mathrm{O}$ due to the current in the larger loop is $B=\dfrac{\mu_{0} I}{2 R}$

If $r$ is the resistance per unit length, then
$I=\dfrac{\text { potential difference }}{\text { resistance }}=\dfrac{4+2.5 t}{2 \pi R \cdot \rho}$
$\therefore \quad B=\dfrac{\mu_{0}}{2 R} \cdot \dfrac{4+2.5 t}{2 \pi R \rho}$
$\therefore r<$\phi=B \times \pi r^{2}=\dfrac{\mu_{0}}{2 R} \cdot \dfrac{4+2.5 t}{2 \pi R \rho} \cdot \pi r^{2}$.

Induced emf $e=\dfrac{d \phi}{d t}=\dfrac{\mu_{0} r^{2}}{4 R^{2} \rho} \times 2.5$

The corresponding current in the smaller loop is $I^{\prime}$ then
$I^{\prime}=\dfrac{e}{R}=\dfrac{\mu_{0} r^{2}}{4 R^{2} \rho} \times 2.5 \times \dfrac{1}{2 \pi r \rho}$
$=\dfrac{2.5{{\mu }_{0}}r}{8\pi {{R}^{2}}{{\rho }^{2}}}=\dfrac{2.5\times 4\pi \times {{10}^{-7}}\times 0.1}{8\pi \times {{(1)}^{2}}\times {{\left( {{10}^{-4}} \right)}^{2}}}$
\[\therefore {{I}^{\prime }}=1.25~\text{A}\]
The current in the smaller loop is \[1.25~\text{A}\]

Therefore, the correct option is (B).

Note:
Around it, the current carrying conductor produces its own magnetic field. ... There will be attraction and repulsion between them when two magnetic fields interact, based on the direction of the external magnetic field and the direction of the current in the conductor. That is why there is a force experienced by the conductor. So, because of this current through this conductor, an electromagnetic field is created. The Maxwell right-hand thumb rule indicates the direction of the current in the direction of the magnetic field.