Cylindrical Capacitors

1. Capacitance of a cylindrical capacitor:

$$ C = \frac{2 \pi \epsilon_0 L}{\ln(b/a)} $$ Where:

• ( C ) is capacitance in Farads (F)
• (\epsilon_0 ) is the permittivity of vacuum ((8.85 \times 10^{-12} \ F/m))
• ( L ) is the length of the capacitor in meters (m)
• ( a ) is the radius of the inner cylinder in meters (m)
• ( b ) is the radius of the outer cylinder in meters (m)

2. Energy stored in a cylindrical capacitor:

$$ U = \frac{1}{2} CV^2 $$

Where:

• ( U ) is the energy stored in Joules (J)
• ( C ) is capacitance in Farads (F)
• ( V ) is the voltage across the capacitor in Volts (V)

3. Parallel and series combinations of cylindrical capacitors:

• Capacitors in Parallel: $$ C_{eq} = C_1 + C_2 $$
• Capacitors in Series: $$ \frac{1}{C_{eq}}= \frac{1}{C_{1}} + \frac{1}{C_2}$$

Spherical Capacitors

1. Capacitance of a spherical capacitor:

$$ C = 4 \pi \epsilon_0 \frac{r_1 r_2}{r_2-r_1} $$ Where:

• ( C ) is capacitance in Farads (F)
• (\epsilon_0 ) is the permittivity of vacuum ((8.85 \times 10^{-12} \ F/m))
• ( r_1 ) is the radius of the inner sphere in meters (m)
• ( r_2 ) is the radius of the outer sphere in meters (m)

2. Energy stored in a spherical capacitor:

$$ U = \frac{1}{2} CV^2 $$ Where:

• ( U ) is the energy stored in Joules (J)
• ( C ) is capacitance in Farads (F)
• ( V ) is the voltage across the capacitor in Volts (V)

3. Parallel and series combinations of spherical capacitors:

• Capacitors in Parallel: $$ C_{eq} = C_1 + C_2 $$
• Capacitors in Series: $$ \frac{1}{C_{eq}}= \frac{1}{C_{1}} + \frac{1}{C_2}$$

Series and Parallel Combinations

1. Equivalent capacitance of capacitors in series and parallel:

• Series : $$ \frac{1}{C_{eq}} = \frac{1}{C_1} + \frac{1}{C_2} + … + \frac{1}{C_n}$$
• Parallel: $$ C_{eq} = C_1 + C_2 + … + C_n$$

2. Voltage division in series capacitors:

$$ V_C_1 = \frac{Q_{C_1}}{C_{C_1}} = \frac{C_{C_2}}{C_{C_1} + C_{C_2}} V_{C} $$ $$ V_C_2 = \frac{Q_{C_2}}{C_{C_2}} = \frac{C_{C_1}}{C_{C_1} + C_{C_2}} V_{C} $$ Where:

• ( Q ) is the charge in Coulombs (C)
• ( C ) is the capacitance in Farads (F)
• ( V ) is the voltage in Volts (V)

3. Charge division in parallel capacitors:

(Q_C = Q_{C_1} =Q_{C_2})