In a Wheatstone bridge with \( R_1 = 30 \, \Omega \), \( R_2 = 60 \, \Omega \), \( R_3 = 15 \, \Omega \), and \( R_4 = 32 \, \Omega \), a \( 10 \, \text{V} \) battery is connected across AC. What is the current through the galvanometer (\( R_g = 10 \, \Omega \))?
Apply Kirchhoff’s rules. Let currents be \( I_1 \) (AB), \( I_2 \) (AD), \( I_g \) (BD).
Junction B: \( I_1 = I_g + I_4 \), Junction D: \( I_2 = I_g + I_3 \).
Loop BADB: \( 30 I_1 + 10 I_g - 60 I_2 = 0 \Rightarrow 3 I_1 + I_g - 6 I_2 = 0 \).
Loop BCDB: \( 60 (I_1 - I_g) - 10 I_g - 15 (I_2 + I_g) = 0 \Rightarrow 4 I_1 - 2 I_2 - 2 I_g = 0 \).
Loop ADCEA: \( 60 I_2 + 15 (I_2 + I_g) = 10 \Rightarrow 75 I_2 + 15 I_g = 10 \Rightarrow 5 I_2 + I_g = \frac{2}{3} \).
Solve: From (3) \( I_g = \frac{2}{3} - 5 I_2 \), substitute in (1): \( 3 I_1 + \frac{2}{3} - 5 I_2 - 6 I_2 = 0 \Rightarrow 3 I_1 - 11 I_2 = -\frac{2}{3} \).
From (2): \( 4 I_1 - 2 I_2 - 2 (\frac{2}{3} - 5 I_2) = 0 \Rightarrow 4 I_1 - 2 I_2 - \frac{4}{3} + 10 I_2 = 0 \Rightarrow 4 I_1 + 8 I_2 = \frac{4}{3} \).
Solve: \( 12 I_1 - 33 I_2 = -2 \), \( 12 I_1 + 24 I_2 = 4 \). Subtract: \( -57 I_2 = -6 \Rightarrow I_2 = \frac{6}{57} = \frac{2}{19} \, \text{A} \).
\( I_g = \frac{2}{3} - 5 \times \frac{2}{19} = \frac{38}{57} - \frac{30}{57} = \frac{8}{57} \approx 0.14 \, \text{A} \).