An 8.3 uF and a 2.9 uF capacitor are connected in series across a 24 V battery. What voltage is required to charge a parallel combination of the two capacitors to the same total energy?
To find the voltage required to charge a parallel combination of the two capacitors to the same total energy, we need to use the formula for the energy stored in a capacitor, which is given by:
E = (1/2) * C * V^2
Where E is the energy, C is the capacitance, and V is the voltage across the capacitor.
First, let's calculate the energy stored in the series combination of the two capacitors:
E_series = (1/2) * (C1 + C2) * V_series^2
Where C1 and C2 are the capacitances of the individual capacitors and V_series is the voltage across the series combination.
Given:
C1 = 8.3 uF (microfarads)
C2 = 2.9 uF (microfarads)
V_series = 24 V
Substituting these values into the equation, we have:
E_series = (1/2) * (8.3 uF + 2.9 uF) * (24 V)^2
Next, let's calculate the energy stored in the parallel combination of the two capacitors:
E_parallel = (1/2) * (C1 * V_parallel^2 + C2 * V_parallel^2)
Where V_parallel is the voltage across the parallel combination.
To make the total energy stored in both cases equal, we set E_series equal to E_parallel:
(1/2) * (8.3 uF + 2.9 uF) * (24 V)^2 = (1/2) * (8.3 uF * V_parallel^2 + 2.9 uF * V_parallel^2)
Now we can solve for V_parallel:
(8.3 uF + 2.9 uF) * (24 V)^2 = 8.3 uF * V_parallel^2 + 2.9 uF * V_parallel^2
(11.2 uF) * (24 V)^2 = (8.3 uF + 2.9 uF) * V_parallel^2
(11.2 uF) * (24 V)^2 / (8.3 uF + 2.9 uF) = V_parallel^2
Taking the square root of both sides, we get:
V_parallel = sqrt((11.2 uF) * (24 V)^2 / (8.3 uF + 2.9 uF))
Now we can calculate the value of V_parallel using a calculator.