Question

derive the expression for resultant capacitor, when capacitors are connected in parallel​

Answer 1

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Answer 2

Answer:

Capacitor in series and parallel: (i) Capacitor in series: Consider three capacitors of capacitance C1 , C2 and C3 connected in series with a battery of voltage V as shown in the figure (a). As soon as the battery is connected to the capacitors in series, the electrons of charge -Q are transferred from negative terminal to the right plate of C3 which pushes the electrons of same amount -Q from left plate of C3 to the right plate of C2 due to electrostatic induction. Similarly, the left plate of C2 pushes the charges of Q to the right plate of which induces the positive charge +Q on the left plate of C1 At the same time, electrons of charge -Q are transferred from left plate of C1 to positive terminal of the battery. By these processes, each capacitor stores the same amount of charge Q. The capacitances of the capacitors are in general different, so that the voltage across each capacitor is also different and are denoted as V1 , V2 and V3 respectively. The total voltage across each capacitor must be equal to the voltage of the a battery. V = V1 + V2 + V3 ….. (1) Since Q = CV, we have V = QC1QC1 + QC2QC2 + QC3QC3 If three capacitors in series are considered to form an equivalent single capacitor Cs shown in figure (b), then we have V = QCsQCs Substituting this expression into equation (2) we get Thus, the inverse of the equivalent capacitance Cs of three capacitors connected in series is equal to the sum of the inverses of each capacitance. This equivalent capacitance Cs is always less than the smallest individual capacitance in the series. (ii) Capacitance in parallel: Consider three capacitors of capacitance C1 ,C2 and C3 connected in parallel with a battery of voltage V as shown in figure (a). 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 Q3 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, If these three capacitors are considered to form a single capacitance C which stores the total charge Q as shown in the figure (b), then we can write Q = CPV. Substituting this in equation (2), we get 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.