We need to understand an important concept: charge is conserved when capacitors are connected without a battery.
Initially, only one capacitor is charged. The charge stored in it can be calculated using the formula:
Charge = Capacitance × Voltage
So, the initial charge is Q = C × V?.
After disconnecting the battery, this charged capacitor is connected to another capacitor. Since there is no battery in the circuit now, no new charge is added or removed. This means the total charge remains the same.
When both capacitors are connected, they reach the same voltage V. Now the total charge is shared between both capacitors:
Charge on first capacitor = C × V
Charge on second capacitor = C? × V
So, total charge becomes:
C × V? = C × V + C? × V
Now, we solve for C?:
First, subtract C × V from both sides:
C(V? − V) = C? × V
Finally, divide both sides by V:
C? = C(V? − V) / V
This formula gives the value of the unknown capacitance.
The initial charge spreads between both capacitors, and by using charge conservation, we can find the unknown capacitance easily.