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(A) In the steady state,the capacitors act as open circuits for $DC$ current. However,the problem states that when the switch is closed,the galvanomet

Vedclass · Accepted Answer

$\frac{R_1}{R_2} = \frac{C_2}{C_1}$

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(A) In the steady state,the capacitors act as open circuits for $DC$ current. However,the problem states that when the switch is closed,the galvanometer shows no deflection. This implies that the potential at the two terminals of the galvanometer is the same (balanced bridge condition).Let the potential of the battery be $V$. The circuit consists of two parallel branches connected to the battery. One branch has $R_1$ and $C_1$ in series,and the other has $R_2$ and $C_2$ in series.Since the galvanometer shows no deflection,the potential at the junction between $R_1$ and $C_1$ must be equal to the potential at the junction between $R_2$ and $C_2$.Let the current flowing through the left branch be $i_1$ and through the right branch be $i_2$. In steady state,the current through the capacitors is zero. However,the question implies a potential divider arrangement where the potential at the nodes is determined by the resistors and capacitors.For the bridge to be balanced,the potential drop across $R_1$ must be equal to the potential drop across $R_2$ relative to the common node,or more simply,the ratio of potentials must be equal. Using the voltage divider rule for the nodes:$V_{node1} = V \cdot \frac{1/C_1}{R_1 + 1/C_1} = V \cdot \frac{1}{1 + R_1 C_1 s}$ (in Laplace domain) or simply considering the steady state potential distribution:The potential at the junction of $R_1$ and $C_1$ is $V_1 = V \frac{1/C_1}{R_1 + 1/C_1}$ is incorrect as they are in series. Actually,the potential at the junction is determined by the voltage divider. For no deflection,the potentials must be equal: $V \frac{1/C_1}{R_1 + 1/C_1} = V \frac{1/C_2}{R_2 + 1/C_2}$.Simplifying this: $\frac{1}{R_1 C_1 + 1} = \frac{1}{R_2 C_2 + 1}$.This implies $R_1 C_1 = R_2 C_2$,which leads to $\frac{R_1}{R_2} = \frac{C_2}{C_1}$.

The circuit shown is in a steady state. Now,when the switch is closed,the galvanometer shows no deflection. Then the correct relation is:

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