The capacitance of a flat capacitor is found from C = ε·ε₀·S/d: S is the area of one plate, d is the distance between the plates, ε is the relative permittivity of the dielectric (about 1 for air), and ε₀ = 8.85·10⁻¹² F/m is the electric constant. So the capacitance grows if the area grows or the distance shrinks, and a dielectric also raises it. For capacitors in parallel the total capacitance is the sum: C = C₁ + C₂; in series 1/C = 1/C₁ + 1/C₂, so the total capacitance is smaller than any single one. Parallel connection is convenient for getting a large capacitance. In series the voltage is shared among the parts, so it is used to withstand a high voltage.
Worked examples
Air (ε = 1), S = 0.02 m², d = 1 mm = 0.001 m: C = 8.85·10⁻¹² · 0.02 : 0.001 = 1.77·10⁻¹⁰ F = 177 pF.
3 µF and 5 µF in parallel: C = 3 + 5 = 8 µF. 6 µF and 3 µF in series: C = 6 · 3 : (6 + 3) = 2 µF.
Class activity
Draw two diagrams in your notebook: two capacitors in parallel and in series. For each write the formula for total capacitance and calculate it for 2 µF and 2 µF (results: 4 µF and 1 µF).
Practice
1
What does the capacitance of a flat capacitor depend on?
On the plate area S, the distance d and the permittivity ε: C = εε₀S/d.
2
What is the total capacitance in µF of 5 µF, 7 µF and 8 µF capacitors in parallel?
20
3
What is the total capacitance in µF of 12 µF and 4 µF capacitors in series?
3
4
Why does the capacitance grow if the distance between the plates is reduced?
In C = εε₀S/d the distance is in the denominator, so smaller d gives larger C.