Q.Two long coaxial solenoids, each of length $0.5\ \text{m}$, are wound one inside the other: the inner solenoid has radius $2\ \text{cm}$ and $1000$ turns per metre, and the outer solenoid has $4000$ turns per metre. Calculate the mutual inductance of the pair.

Given: $l=0.5\ \text{m}$ (both solenoids), inner radius $r_1=2\ \text{cm}=0.02\ \text{m}$, $n_1=1000\ \text{turns/m}$ (inner), $n_2=4000\ \text{turns/m}$ (outer).

Step 1 -- $n_1 n_2$:

Step 2 -- $\pi r_1^2$:

Step 3 -- combine with $\mu_0$, $l$:

$1.2566\times10^{-6}\times4\times10^{6} = 5.0265$ …

Apply $M=\mu_0n_1n_2\pi r_1^2l$ using the inner (smaller) radius $r_1$, working t …

• Using the outer solenoid's radius instead of the inner one -- the formula always uses the SMALLER radius, regardless of which solenoid's turn density is written first. …