A tunable capacitor (with variable capacitance) is charged by a U
0=12 V battery and then is connected in parallel to
a R=3 Ω resistor. The capacitance
C(t) of the capacitor
is controlled so that the current in the circuit remains constant at all times. What is the power (external) in Watts needed to keep the current in the circuit constant? Note that one possible way of controlling the capacitance could be varying the distance between the plates of the capacitor which, in general, requires some work.
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To find the power (external) required to keep the current constant in the circuit, we need to first understand the behavior of a capacitor in a DC circuit.
When a capacitor is connected in parallel to a resistor in a DC circuit, it starts to discharge, and the current flowing through the circuit gradually decreases over time. However, if the capacitance is controlled in such a way that the current remains constant, it implies that the capacitor is being charged/recharged continuously to counterbalance the discharge.
In this case, as the capacitor is being charged by a 12V battery, the voltage across the capacitor remains constant. Let's assume this voltage as Vc.
The instantaneous power (external) required in the circuit can be calculated using the formula:
P(t) = I(t) * V(t)
Where:
- P(t) is the power (external) at any instant of time t.
- I(t) is the current at that particular time t.
- V(t) is the voltage across the resistor at time t.
Since the current is constant in this circuit, we denote it as I.
Now, we need to determine the voltage across the resistor when the capacitor is connected. This voltage can be found using Ohm's Law:
V(t) = I * R
Substituting the value of V(t) in the power formula, we get:
P(t) = I * (I * R) = I^2 * R
So, the external power needed to keep the current constant is given by P = I^2 * R.
In this case, as the current is constant, we can calculate I by dividing the initial charge on the capacitor by the time it takes to fully discharge.
Let's assume the initial charge on the capacitor is Q0 and the time taken to fully discharge is t0.
I = Q0 / t0
Substituting the value of I in the power formula, we have:
P = (Q0 / t0)^2 * R
Please note that the equation assumes that the time for the capacitor to discharge fully is much longer than the time required to charge it. This ensures that the capacitor is continuously charged/recharged to maintain a constant current.
To calculate the exact value, we would need the specific values of charge and time.