Question 82: Electrostats & Capacitance

Question

A circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is

Accepted Answer

Step 1: Initial State (Switch Open)
Let $q_1$ and $V_1$ be the charge and voltage on capacitor $C_1 = 3\,\mu\mathrm{F}$. Let $q_2$ and $V_2$ be the charge and voltage on capacitor $C_2 = 6\,\mu\mathrm{F}$.
When the switch $S$ is open, the two capacitors are connected in series. This means that the charge stored on each capacitor is the same. So, $q_1 = q_2$.
The voltages on the capacitors add up to the battery voltage:

$$V_1 + V_2 = 9\,\mathrm{V}$$
The charges and voltages on capacitors in series also follow this equation:

$$C_1 V_1 = C_2 V_2$$

This means the charge on each capacitor is the same since $q_1 = C_1 V_1$ and $q_2 = C_2 V_2$.
Solving these two equations gives:
$V_1 = 6\,\mathrm{V}$, $q_1 = 3\,\mu\mathrm{F} 	imes 6\,\mathrm{V} = 18\,\mu\mathrm{C}$
$V_2 = 3\,\mathrm{V}$, $q_2 = 6\,\mu\mathrm{F} 	imes 3\,\mathrm{V} = 18\,\mu\mathrm{C}$
Total charge on the side connected to $X$ is $-q_1 + q_2 = 0$. No net charge has moved yet.
Step 2: After Closing the Switch
When the switch is closed, the resistors $R_1 = 3\,\Omega$ and $R_2 = 6\,\Omega$ are connected in series with the capacitors and battery.
Total resistance is $R_1 + R_2 = 9\,\Omega$.
The current from the battery is:

$$I = \frac{9\,\mathrm{V}}{9\,\Omega} = 1\,\mathrm{A}$$
The voltage drop across each resistor is:
$V_1' = I 	imes R_1 = 1\,\mathrm{A} 	imes 3\,\Omega = 3\,\mathrm{V}$
$V_2' = I 	imes R_2 = 1\,\mathrm{A} 	imes 6\,\Omega = 6\,\mathrm{V}$
The new charges on the capacitors are:
$q_1' = C_1 	imes V_1' = 3\,\mu\mathrm{F} 	imes 3\,\mathrm{V} = 9\,\mu\mathrm{C}$
$q_2' = C_2 	imes V_2' = 6\,\mu\mathrm{F} 	imes 6\,\mathrm{V} = 36\,\mu\mathrm{C}$
Final Calculation
The total charge on the plates connected to $X$ is now $-q_1' + q_2' = -9 + 36 = 27\,\mu\mathrm{C}$.
This means that $27\,\mu\mathrm{C}$ of charge flows from $Y$ to $X$ when the switch is closed.

Answer

0

Answer

54 $$\mu$$C

Answer

27 $$\mu$$C

Answer

81 $$\mu$$C

Electrostats & Capacitance

A circuit is connected as shown in the figure with the switch S open. When the switch is closed, the total amount of charge that flows from Y to X is

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