Two co-axial solenoids having number of turns per unit length 50 and 200, respectively have same length 50 cm. The area of cross-section of the two solenoids are \( 4\,\mathrm{cm^2} \) and \( 6\,\mathrm{cm^2} \), respectively. The mutual inductance of the two solenoids is
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → \( 8\pi \times 10^{-7}\,\mathrm{H} \)
For two long co-axial solenoids of the same length \( l \), the field of the outer solenoid fills the inner one, and the flux is linked only through the cross-section of the inner solenoid (area \( A_1 \), the smaller one).
\( \displaystyle M = \mu_0\,n_1 n_2\,A_1\,l \)
\( n_1 = 50\,\mathrm{m^{-1}} \)
\( n_2 = 200\,\mathrm{m^{-1}} \)
\( A_1 = 4\,\mathrm{cm^2} = 4\times10^{-4}\,\mathrm{m^2} \)
\( l = 50\,\mathrm{cm} = 0.5\,\mathrm{m} \)
\( \displaystyle M = 4\pi\times10^{-7} \times 50 \times 200 \times 4\times10^{-4} \times 0.5 \)
\( \displaystyle M = 4\pi\times10^{-7} \times 10^{4} \times 2\times10^{-4} = 8\pi\times10^{-7}\,\mathrm{H} \)