Question 14

The quantity $\sqrt{\mu_0/\epsilon_0}$ is the impedance of free space, which has dimensions of resistance ($[R]$). The dimensions of resistance are $[R] = [ML^2 T^{-3} I^{-2}]$. Let the magnitude in the new system be $n_2$. Using $n_1 u_1 = n_2 u_2$: $1 \cdot [M_1 L_1^2 T_1^{-3} I_1^{-2}] = n_2 \cdot [M_2 L_2^2 T_2^{-3} I_2^{-2}]$ $n_2 = 1 \cdot \left(\frac{M_1}{M_2}\right)^1 \cdot \left(\frac{L_1}{L_2}\right)^2 \cdot \left(\frac{T_1}{T_2}\right)^{-3} \cdot \left(\frac{I_1}{I_2}\right)^{-2}$ Substituting the given units ($M_2 = 5 M_1$, $L_2 = 5 L_1$, $T_2 = 5 T_1$, $I_2 = 5 I_1$): $n_2 = 1 \cdot \left(\frac{1}{5}\right) \cdot \left(\frac{1}{5}\right)^2 \cdot \left(\frac{1}{5}\right)^{-3} \cdot \left(\frac{1}{5}\right)^{-2}$ $n_2 = 5^{-1} \cdot 5^{-2} \cdot 5^3 \cdot 5^2 = 5^2 = 25$.