Capacitors in Series Derivation

Here is a derivation for two electrolytic capacitors in series. The diagram shows how to connect the electrolytic capacitors, where the positive terminal joins to the negative terminal. The goal is to prove the formula for capacitors in series, or equivalent capacitance. After this, you can write your own lab report.

The interesting thing about this configuration occurs at the centre plates. One plate has a positive charge, whilst another plate has a negative charge, and an electrical conductor connects them together. Since they are electrically bonded, the net charge across the inner plates will be zero. Since the charges there are equal and opposite, it follows that the charges at their opposing plates must also be equal and opposite, which is why both capacitors have the same charge.

This describes a voltage V1 across capacitor C1, with charge q1.

This describes a voltage V2 across capacitor C2, with charge q2.

This is the expression for total voltage of the system.

This expression describes the voltage across capacitors in series. Whether it is Kirchhoff’s rule or common sense, the voltage Vtot must be equal to the sum of voltages V1 and V2. Substituting the previous expressions into this equation gives us the following.

This is the same as the previous expression but in terms of charge and capacitance, and thus eliminating the voltage terms from the equation.

The charge for capacitors in series is the same, therefore qtot = q1 = q2. Capacitors in series share the same charge because the charge comes from the neighbouring plate. The total charge is equal to q1 and q2, and therefore the charge is equal. Knowing this, the charge terms cancel out by dividing the previous expression throughout by q to give the following expression.

This reciprocal formula gives the total capacitance.

Capacitors in Series Formula

After rearranging, we get this formula for capacitors in series.

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