Capacitance Shortcut Methods and Tricks

1. Cylindrical Capacitor

• Capacitance of a cylindrical capacitor:

$$C = \frac{2\pi\varepsilon_0 l}{\ln \frac{b}{a}}$$

• Calculate the natural logarithm of the ratio of outer to inner radius: $$ \ln \frac{b}{a}$$
• Multiply this value by the length of the capacitor: $$ l\times\ln \frac{b}{a} $$
• Multiply the result (from the previous step) by the permittivity of free space, (\varepsilon_0)$: $$ \varepsilon_0\times l\times\ln \frac{b}{a} $$
• Multiply the final value by $$ 2\pi\times \varepsilon_0\times l\times\ln \frac{b}{a} $$

2. Spherical Capacitor

• Capacitance of a spherical capacitor: $$C = 4\pi\varepsilon_0 \frac{ab}{b-a}$$

• Calculate the product of the inner and outer radii: $$ ab$$

• Subtract the inner radius from the outer radius: $$ b-a$$

• Divide the product of the radii from the previous step by the difference in radii: $$\frac{ab}{b-a}$$

• Multiply this value by the permittivity of free space, (\varepsilon_0)$: $$ \varepsilon_0\times \frac{ab}{b-a}$$

• Finally, multiply the result by (4\pi\times\varepsilon_0): $$4\pi\varepsilon_0\times \frac{ab}{b-a}$$

3. Series Combination of Capacitors

• Capacitance of capacitors in series: $$ \frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots $$

• Calculate the reciprocal of each individual capacitance.

• Add these reciprocals together.

• Calculate the reciprocal of the sum to find the total capacitance.

4. Parallel Combination of Capacitors

• Capacitance of capacitors in parallel: $$C_{parallel} = C_1 + C_2 + C_3 +\dots$$

• Simply add up the capacitance values of all capacitors in parallel to find the total capacitance.

5. Cylindrical vs. Spherical Capacitance Comparison

• Ratio of cylindrical capacitance to spherical capacitance: $$\frac{C_{cylindrical}}{C_{spherical}} = \frac{\ln \frac{b}{a}}{\frac{b}{b-a}}$$

• Calculate the natural logarithm of the ratio of outer to inner radius of cylindrical capacitor: $$ \ln \frac{b}{a} $$

• Calculate the radius ratio $$\frac{b}{b-a} $$

• Divide the logarithmic value of cylindrical capacitor by the radius ratio of the spherical capacitor.