Two capacitors of unknown capacitances c1 and c2 are connected first in series and then in parallel across a battery of 100 v. if the energy stored in the two combinations are 0.045 j and 0.25 j respectively, determine the value of c1 and c2

General
1

two capacitors of unknown capacitances c1 and c2 are connected first in series and then in parallel across a battery of 100 v. if the energy stored in the two combinations are 0.045 j and 0.25 j respectively, determine the value of c1 and c2. also, calculate the charge on each capacitor in a parallel combination.

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What is the solution to the capacitor problem presented by LectureNotes?

Solution:
Given:

  • Capacitance of the first capacitor (C1) is c1.
  • Capacitance of the second capacitor (C2) is c2.
  • Battery voltage applied across the capacitors is 100 V.
  • Energy stored in the series combination (E_series) = 0.045 J.
  • Energy stored in the parallel combination (E_parallel) = 0.25 J.

We know that the energy stored in a capacitor is given by the formula:
[ E = \frac{1}{2}CV^2 ]
where:

  • E is the energy stored,
  • C is the capacitance,
  • V is the voltage applied.

1. Calculating the Capacitance Values (C1 and C2):
For capacitors in series, the total capacitance (Cs) is given by:
[ Cs = \frac{1}{\frac{1}{c1} + \frac{1}{c2}} ]
Given that energy stored in the series combination is 0.045 J:
[ Cs = \frac{1}{2} \times Cs \times V^2 = 0.045 ]
Substitute V = 100 V:
[ \frac{1}{2} \times Cs \times (100)^2 = 0.045 ]
[ Cs = \frac{0.045 \times 2}{100^2} ]
[ Cs = \frac{0.09}{10000} ]
[ Cs = 9 \times 10^{-6} F ]

For capacitors in parallel, the total energy stored (Ep) is the sum of energies of each capacitor:
[ Ep = E1 + E2 = \frac{1}{2}c1V^2 + \frac{1}{2}c2V^2 ]
Given that the energy stored in the parallel combination is 0.25 J:
[ \frac{1}{2}c1V^2 + \frac{1}{2}c2V^2 = 0.25 ]
[ c1 + c2 = \frac{0.25}{V^2} ]
[ c1 + c2 = \frac{0.25}{100^2} ]
[ c1 + c2 = \frac{0.25}{10000} ]
[ c1 + c2 = 2.5 \times 10^{-5} F ]

Solving the equations simultaneously, we find the values of c1 and c2.

2. Calculating the Charge on Each Capacitor in the Parallel Combination:
The charge on each capacitor in parallel can be calculated using the formula:
[ Q = CV ]
For capacitor 1:
[ Q1 = c1 \times V = c1 \times 100 ]
For capacitor 2:
[ Q2 = c2 \times V = c2 \times 100 ]

By substituting the values of c1 and c2 obtained in the previous calculations, we can find the charges on each capacitor in the parallel combination.