Cylindrical And Spherical Capacitors, Series And Parallel Combinations

Slide 1: Cylindrical and Spherical Capacitors

Capacitors are electronic components used to store and release electrical energy.
Cylindrical and spherical capacitors are two common types of capacitors.
They consist of two conductive plates separated by a dielectric material.
The shape and arrangement of the plates determine the behavior of the capacitor.
Cylindrical capacitors have a cylindrical arrangement of plates, while spherical capacitors have a spherical arrangement.

Slide 2: Capacitance of a Cylindrical Capacitor

The capacitance of a cylindrical capacitor is given by the formula: $C = \frac{2\pi\epsilon_0L}{log(\frac{R_2}{R_1})}$
Where:
- C is the capacitance of the capacitor.
- $epsilon_0$ is the permittivity of free space.
- L is the length of the cylinder.
- $R_1$ and $R_2$ are the radii of the inner and outer cylinders, respectively.

Slide 3: Capacitance of a Spherical Capacitor

The capacitance of a spherical capacitor is given by the formula: $C = 4\pi\epsilon_0(\frac{R_1R_2}{R_2-R_1})$
Where:
- C is the capacitance of the capacitor.
- $epsilon_0$ is the permittivity of free space.
- $R_1$ and $R_2$ are the radii of the inner and outer spheres, respectively.

Slide 4: Series Combination of Capacitors

In a series combination of capacitors, the capacitors are connected end to end.
The total capacitance, $C_{\\text{total}}$ , of the series combination is given by the formula: $\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \ldots + \frac{1}{C_n}$
Where:
- $C_1, C_2, C_3, \\ldots, C_n$ are the capacitances of the individual capacitors.

Slide 5: Parallel Combination of Capacitors

In a parallel combination of capacitors, the positive terminals are connected together, and the negative terminals are connected together.
The total capacitance, $C_{\\text{total}}$ , of the parallel combination is given by the formula: $C_{\text{total}} = C_1 + C_2 + C_3 + \ldots + C_n$
Where:
- $C_1, C_2, C_3, \\ldots, C_n$ are the capacitances of the individual capacitors.

Slide 6: Example - Series Combination of Capacitors

Let’s consider the following series combination of capacitors:
- Capacitor 1 with capacitance $C_1$ = 5 µF
- Capacitor 2 with capacitance $C_2$ = 10 µF
- Capacitor 3 with capacitance $C_3$ = 15 µF
To find the total capacitance, we use the formula: $\frac{1}{C_{\text{total}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3}$
Substituting the values, we get: $\frac{1}{C_{\text{total}}} = \frac{1}{5 \times 10^{-6}} + \frac{1}{10 \times 10^{-6}} + \frac{1}{15 \times 10^{-6}}$

Slide 7: Example - Series Combination of Capacitors (Contd.)

Solving the equation, we find: $\frac{1}{C_{\text{total}}} = \frac{1}{(100 + 150 + 200) \times 10^{-6}}$
Simplifying, we get: $\frac{1}{C_{\text{total}}} = \frac{1}{450 \times 10^{-6}}$
Therefore, the total capacitance is: $C_{\text{total}} = \frac{450 \times 10^{-6}}{1} = 2.222 \mu\F$

Slide 8: Example - Parallel Combination of Capacitors

Let’s consider the following parallel combination of capacitors:
- Capacitor 1 with capacitance $C_1$ = 5 µF
- Capacitor 2 with capacitance $C_2$ = 10 µF
- Capacitor 3 with capacitance $C_3$ = 15 µF
To find the total capacitance, we simply add the individual capacitances: $C_{\text{total}} = C_1 + C_2 + C_3$
Substituting the values, we get: $C_{\text{total}} = 5 \times 10^{-6} + 10 \times 10^{-6} + 15 \times 10^{-6}$

Slide 9: Example - Parallel Combination of Capacitors (Contd.)

Solving the equation, we find: $C_{\text{total}} = 25 \times 10^{-6}$
Therefore, the total capacitance is: $C_{\text{total}} = 25 \mu\F$

Slide 10: Summary

Cylindrical and spherical capacitors are two common types of capacitors.
The capacitance of a cylindrical capacitor depends on its length and radii.
The capacitance of a spherical capacitor depends on its radii.
In a series combination of capacitors, the total capacitance is given by the reciprocal sum of individual capacitances.
In a parallel combination of capacitors, the total capacitance is the sum of individual capacitances. Sure! Here are slides 11 to 20:

Slide 11: Electric Potential Energy of a Capacitor

Capacitors store electrical energy in the form of electric potential energy.
The electric potential energy, U, of a charged capacitor is given by the formula:
- Where:
  - U is the electric potential energy
  - Q is the charge on the capacitor plates
  - V is the potential difference between the plates
The energy stored in a capacitor increases as either the charge or the potential difference across the plates increases.

Slide 12: Energy Density of a Capacitor

The energy density, u, of a capacitor is the energy stored per unit volume.
It is given by the formula:
- Where:
  - u is the energy density
  - U is the electric potential energy
  - V is the volume of the capacitor
Energy density provides information about how concentrated the energy is within a given volume.

Slide 13: Dielectric Materials

Dielectric materials are insulators placed between the plates of a capacitor.
They increase the capacitance of the capacitor by reducing the electric field between the plates.
Common dielectric materials include:
- Air
- Paper
- Glass
- Plastics
Dielectric materials have a dielectric constant, ε.
The capacitance with a dielectric material is given by the formula:
- Where:
  - C is the capacitance
  - ε is the dielectric constant
  - ε0 is the permittivity of free space
  - A is the area of the plates
  - d is the distance between the plates

Slide 14: Dielectric Strength

Dielectric strength is the maximum electric field a dielectric material can withstand before breaking down and becoming conductive.
It is a measure of the dielectric’s resistance to electrical breakdown.
Dielectric strength is commonly measured in volts per meter (V/m).
Higher dielectric strength indicates a more robust and reliable dielectric material.

Slide 15: Example - Capacitance with a Dielectric Material

Let’s consider a capacitor with a dielectric material of air.
The area of the plates is 0.01 m² and the distance between the plates is 0.001 m.
The dielectric constant of air is approximately 1.
The permittivity of free space is $epsilon_0$ = 8.854 x 10⁻¹² F/m.
Substituting the values into the capacitance formula, we get: $C = (1) ((8.854 \times 10^{-12})((0.01) / (0.001)))$
Solving the equation, we find: $C = 88.54 \times 10^{-12} \text{F}$

Slide 16: Capacitance of a Parallel Plate Capacitor

A parallel plate capacitor consists of two parallel conductive plates separated by a distance, d.
The capacitance of a parallel plate capacitor is given by the formula:
- Where:
  - C is the capacitance
  - ε0 is the permittivity of free space
  - A is the area of the plates
  - d is the distance between the plates
The capacitance of a parallel plate capacitor increases with the plate area and decreases with the distance between the plates.

Slide 17: Energy Stored in a Parallel Plate Capacitor

The energy stored in a charged parallel plate capacitor is given by the formula:
- Where:
  - U is the energy stored
  - ε0 is the permittivity of free space
  - A is the area of the plates
  - d is the distance between the plates
  - V is the potential difference across the plates
The energy stored in a parallel plate capacitor increases with the plate area, distance between the plates, and the potential difference.

Slide 18: Charging and Discharging a Capacitor

Charging a capacitor:
- When a capacitor is connected to a voltage source, it charges up gradually.
- The charge on the plates increases, and the potential difference across the plates approaches the voltage of the source.
Discharging a capacitor:
- When a charged capacitor is disconnected from the voltage source, it discharges gradually.
- The charge on the plates decreases, and the potential difference across the plates approaches zero.
The charging and discharging processes can be described using exponential functions.

Slide 19: Time Constant of a Discharging Capacitor

The time constant, τ, of a discharging capacitor is a measure of how quickly it discharges.
It is given by the formula:
- Where:
  - τ is the time constant
  - R is the resistance in the discharge circuit
  - C is the capacitance of the capacitor
The time constant represents the time it takes for the charge on the capacitor to decrease to approximately 37% of its initial value.

Slide 20: Applications of Capacitors

Capacitors have various applications in electronic circuits, including:
- Energy storage in power supplies
- Timing circuits in oscillators