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Question
Derive the expression for resultant capacitance, when the capacitor is connected in parallel.
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Solution
Consider three capacitors of capacitance C1,C2 and C3 connected in parallel with a battery of voltage V as shown in figure (a).
(a) capacitors in parallel-
(b) equivalent capacitance with the same total charge
Since corresponding sides of the capacitors are connected to the same positive and negative terminals of the battery, the voltage across each capacitor is equal to the battery’s voltage. Since capacitance of the capacitors is different, the charge stored in each capacitor is not the same. Let the charge stored in the three capacitors be Q1,Q2, and Q2 respectively. According to the law of conservation of total charge, the sum of these three charges is equal to the charge Q transferred by the battery,
Q = Q1 + Q2 + Q3 ….. (1)
Now, since Q = CV, we have
Q = C1V + C2 V + C3 V ….. (2)
If these three capacitors are considered to form a single capacitance CP which stores the total charge Q as shown in figure (b), then we can write Q = CPV. Substituting this in equation (2), we get
Cp V = C1 V + C2 V + C3 V
Cp = C1 + C2 + C3
Thus, the equivalent capacitance of capacitors connected in parallel is equal to the sum of the individual capacitance. The equivalent capacitance Cp in a parallel connection is always greater than the largest individual capacitance. In a parallel connection, it is equivalent as area of each capacitance adds to give more effective area such that total capacitance increases.
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