Optimal turn count for fixed winding volume ($R_{\mathrm{coil}}=R_s$)

For fixed mean turn length and fixed conductor volume, a coil of $N$ turns has conductor length proportional to $N$ and cross-sectional area proportional to $1/N$. Its electrical resistance is therefore $R_{\mathrm{coil}}=a{N}^2$. With source electromotive force $\mathcal{E}$ and internal resistance $R_s$, suppose the small-deflection driving torque is proportional to $NI=N\mathcal{E}/(R_s+a{N}^2)$. Its derivative has the sign of $R_s-a{N}^2$, so the maximum occurs at $R_{\mathrm{coil}}=R_s$. Geometry, damping and finite response time can alter the practical optimization; this is a conditional winding model, not a universal rule for all galvanometers.