With your understanding of concept of Inductance, answer the questions.
An electric current can be induced in a coil by flux change produced by another coil in its vicinity or flux change produced by the same coil. The flux through a coil is proportional to the current i.e \( \phi_{B} \propto I \). If the geometry of the coil does not vary with tune then \( \displaystyle\frac{d\phi}{dt} \propto \frac{dI}{dt} \). For a closely wound coil of N turns, the same magnetic flux. is linked with all turns: i.e \( N\phi_{B} \propto I \). The constant of proportionality is called inductance which depends on geometry of coil and dielectric constant of intervining medium. Mutual inductance of a pain of coil solenoids, etc depends on their separation and relative orientation self-inductance of single coil or solenoid depends on geometry of the coil and permeability of the medium. If plays role of inertia.
A circular coil of radius \( r_2 = 6\,\mathrm{cm} \) has another concentric coil of radius \( r_1 = 0.2\,\mathrm{cm} \) placed inside it. The arrangement produced mutual inductance \( M \). If the inner coil is replaced with another one having radius \( r' = 0.3\,\mathrm{cm} \), select the current statement giving the new value of mutual inductance \( m' \) with respect to the earlier value \( m \).
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( m' = 2.25\,M \)
The inner coil is very small \( (r_1 \ll r_2) \), so the magnetic field of the outer coil can be taken as uniform over it and equal to the field at the centre. The mutual inductance is the flux through the inner coil per unit current in the outer coil.
\( \displaystyle B = \frac{\mu_0 I}{2 r_2} \)
\( \displaystyle \phi = B\,\pi r_1^{2} = \frac{\mu_0 \pi r_1^{2}}{2 r_2}\,I \;\Rightarrow\; M = \frac{\phi}{I} = \frac{\mu_0 \pi r_1^{2}}{2 r_2} \)
The outer coil is unchanged, so \( M \propto r_1^{2} \).
\( \displaystyle \frac{m'}{M} = \left(\frac{0.3\,\mathrm{cm}}{0.2\,\mathrm{cm}}\right)^{2} = (1.5)^{2} = 2.25 \)
So \( m' = 2.25\,M \).