Two tangent galvanometers, which are identical except in their number of turns, are connected in parallel. The ratio of their resistances of the coils is 1 : 3 . If the deflections in the two tangent galvanometers are \(30^{\circ}\) and \(60^{\circ}\) respectively, then the ratio of their number of turns is

We know that the current passing through galvanometer coil is
\(I=K \tan \theta\) where \(K=\frac{2 r B_{H}}{n \mu_{0}}\)
\(\therefore\) Here, \( I_{1}=K_{1} \tan \theta_{1}\) and \(I_{2}=K_{2} \tan \theta_{2}\)
or \( \frac{l_{1}}{I_{2}}=\frac{K_{1} \tan \theta_{1}}{K_{2} \tan \theta_{2}}=\frac{\frac{2 r B_{H}}{n_{1} \mu_{0}} \tan \theta_{1}}{\frac{2 r B_{H}}{n_{2} \mu_{0}} \tan \theta_{2}}\)
\(\Rightarrow \frac{R_{2}}{R_{1}}=\frac{n_{2} \tan 30^{\circ}}{n_{1} \tan 60^{\circ}}\)
\(\Rightarrow \frac{3}{1}=\frac{n_{2}}{n_{1}} \cdot \frac{\sqrt{3}}{\sqrt{3}} \left(\because I=\frac{V}{R}\right)\)
\(\Rightarrow \frac{n_{1}}{n_{2}}=\frac{1}{3}\)