### Cylindrical And Spherical Capacitors Series And Parallel Combinations

**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})