Cylindrical and Spherical Capacitors

  • Capacitors are devices that store electric charge.
  • Cylindrical and spherical capacitors are two types of capacitors with different shapes.

Cylindrical Capacitors

  • Cylindrical capacitors consist of two coaxial cylinders.
  • The inner cylinder acts as the positive plate, and the outer cylinder acts as the negative plate.
  • The space between the cylinders is filled with a dielectric material.

Spherical Capacitors

  • Spherical capacitors consist of two concentric spheres.
  • The inner sphere acts as the positive plate, and the outer sphere acts as the negative plate.
  • The space between the spheres is filled with a dielectric material.

Series Combination of Cylindrical Capacitors

  • In a series combination, the positive plate of one capacitor is connected to the negative plate of the next capacitor.
  • The total capacitance of the series combination is given by the reciprocal of the sum of the reciprocals of individual capacitances.

Parallel Combination of Cylindrical Capacitors

  • In a parallel combination, the positive plates of all capacitors are connected together, and the negative plates are connected together.
  • The total capacitance of the parallel combination is the sum of individual capacitances.

Series Combination of Spherical Capacitors

  • In a series combination, the positive plate of one capacitor is connected to the negative plate of the next capacitor.
  • The total capacitance of the series combination is given by the reciprocal of the sum of the reciprocals of individual capacitances.

Parallel Combination of Spherical Capacitors

  • In a parallel combination, the positive plates of all capacitors are connected together, and the negative plates are connected together.
  • The total capacitance of the parallel combination is the sum of individual capacitances.

Example - Series Combination of Cylindrical Capacitors

  • Let’s consider two cylindrical capacitors with capacitances C1 and C2.
  • The total capacitance of the series combination is given by (1/C) = (1/C1) + (1/C2).
  • For example, if C1 = 5 μF and C2 = 3 μF, then the total capacitance is C = 1/[(1/5) + (1/3)] = 1.875 μF.

Example - Parallel Combination of Cylindrical Capacitors

  • Let’s consider two cylindrical capacitors with capacitances C1 and C2.
  • The total capacitance of the parallel combination is given by C = C1 + C2.
  • For example, if C1 = 5 μF and C2 = 3 μF, then the total capacitance is C = 5 μF + 3 μF = 8 μF.

Example - Series Combination of Spherical Capacitors

  • Let’s consider two spherical capacitors with capacitances C1 and C2.
  • The total capacitance of the series combination is given by (1/C) = (1/C1) + (1/C2).
  • For example, if C1 = 5 μF and C2 = 3 μF, then the total capacitance is C = 1/[(1/5) + (1/3)] = 1.875 μF. '

Slide 11: Capacitors - A Brief Recap

  • Capacitors are electrical devices used to store and release electrical energy.
  • They consist of two conductive plates separated by a dielectric material.
  • The capacitance of a capacitor represents its ability to store charge and is given by the equation: C = Q/V, where C is the capacitance, Q is the charge stored, and V is the potential difference across the plates.

Slide 12: Series Combination - Capacitors

  • In a series combination of capacitors, the positive plate of one capacitor is connected to the negative plate of the next capacitor.
  • The total capacitance of the series combination is given by the reciprocal of the sum of the reciprocals of individual capacitances: (1/C_total) = (1/C1) + (1/C2) + …

Slide 13: Example - Series Combination

  • Let’s consider two capacitors with capacitances C1 and C2 connected in series.
  • The total capacitance is given by (1/C_total) = (1/C1) + (1/C2).
  • For example, if C1 = 10 μF and C2 = 20 μF, then the total capacitance is C_total = 1/[(1/10) + (1/20)] = 6.6 μF.

Slide 14: Parallel Combination - Capacitors

  • In a parallel combination of capacitors, the positive plates of all capacitors are connected together, and the negative plates are connected together.
  • The total capacitance of the parallel combination is the sum of individual capacitances: C_total = C1 + C2 + …

Slide 15: Example - Parallel Combination

  • Let’s consider two capacitors with capacitances C1 and C2 connected in parallel.
  • The total capacitance is given by C_total = C1 + C2.
  • For example, if C1 = 10 μF and C2 = 20 μF, then the total capacitance is C_total = 10 μF + 20 μF = 30 μF.

Slide 16: Cylindrical Capacitors - Introduction

  • Cylindrical capacitors consist of two coaxial cylinders.
  • The inner cylinder acts as the positive plate, and the outer cylinder acts as the negative plate.
  • The space between the cylinders is filled with a dielectric material.

Slide 17: Spherical Capacitors - Introduction

  • Spherical capacitors consist of two concentric spheres.
  • The inner sphere acts as the positive plate, and the outer sphere acts as the negative plate.
  • The space between the spheres is filled with a dielectric material.

Slide 18: Capacitance of Cylindrical Capacitors

  • The capacitance of a cylindrical capacitor is given by the equation: C = (2πε₀L) / ln(b/a), where C is the capacitance, ε₀ is the permittivity of free space, L is the length of the cylinder, a is the radius of the inner cylinder, and b is the radius of the outer cylinder.

Slide 19: Example - Capacitance of Cylindrical Capacitors

  • Let’s consider a cylindrical capacitor with a length L = 10 cm, inner cylinder radius a = 2 cm, and outer cylinder radius b = 5 cm.
  • The capacitance is given by C = (2πε₀L) / ln(b/a).
  • By substituting the values, the capacitance is: C = (2π * (8.85 x 10^-12 F/m) * 0.1 m) / ln(5/2) = 1.61 x 10^-10 F.

Slide 20: Capacitance of Spherical Capacitors

  • The capacitance of a spherical capacitor is given by the equation: C = (4πε₀ab) / (b - a), where C is the capacitance, ε₀ is the permittivity of free space, a is the radius of the inner sphere, and b is the radius of the outer sphere.

Slide 21: Capacitance of Spherical Capacitors (Continued)

  • The capacitance of a spherical capacitor is given by the equation: C = (4πε₀ab) / (b - a), where C is the capacitance, ε₀ is the permittivity of free space, a is the radius of the inner sphere, and b is the radius of the outer sphere.

Slide 22: Example - Capacitance of Spherical Capacitors

  • Let’s consider a spherical capacitor with an inner sphere radius a = 3 cm and outer sphere radius b = 6 cm.
  • The capacitance is given by C = (4πε₀ab) / (b - a).
  • By substituting the values, the capacitance is: C = (4π * (8.85 x 10^-12 F/m) * 0.03 m * 0.06 m) / (0.06 m - 0.03 m) = 3.75 x 10^-11 F.

Slide 23: Dielectric Material

  • Dielectric materials are non-conductive substances used to separate the plates of a capacitor.
  • They increase the capacitance by reducing the electric field and increasing the charge storage.
  • Dielectric materials have a high relative permittivity compared to vacuum or air.

Slide 24: Relative Permittivity

  • Relative permittivity (εᵣ) is a measure of how much a dielectric material can increase the capacitance compared to vacuum or air.
  • It is defined as the ratio of the capacitance with the dielectric material (C) to the capacitance without any dielectric material (C₀): εᵣ = C / C₀.
  • The relative permittivity value varies for different dielectric materials.

Slide 25: Capacitance with Dielectric Material

  • The capacitance of a capacitor with a dielectric material is given by the equation: C = εᵣ * C₀, where C is the capacitance, εᵣ is the relative permittivity, and C₀ is the capacitance without any dielectric material (in vacuum or air).

Slide 26: Example - Capacitance with Dielectric Material

  • Let’s consider a capacitor without any dielectric material (in air) with a capacitance C₀ = 5 μF.
  • Now, let’s introduce a dielectric material with a relative permittivity εᵣ = 4.
  • The capacitance with the dielectric material is given by C = εᵣ * C₀ = 4 * 5 μF = 20 μF.

Slide 27: Energy Stored in a Capacitor

  • A capacitor stores electrical energy in its electric field.
  • The energy stored in a capacitor is given by the equation: U = (1/2) * C * V², where U is the energy stored, C is the capacitance, and V is the potential difference across the plates.

Slide 28: Example - Energy Stored in a Capacitor

  • Let’s consider a capacitor with a capacitance C = 10 μF and a potential difference V = 50 V.
  • The energy stored in the capacitor is given by U = (1/2) * C * V².
  • By substituting the values, the energy stored is: U = (1/2) * 10 μF * (50 V)² = 12.5 mJ.

Slide 29: Charging and Discharging of Capacitors

  • When a capacitor is connected to a power source, it charges up until it reaches the same potential as the source.
  • The charging process is exponential and follows the equation: V = V₀ * (1 - e^(-t/RC)), where V is the potential difference at time t, V₀ is the initial potential difference, R is the resistance in the circuit, and C is the capacitance.
  • When a charged capacitor is disconnected from the power source, it discharges over time.
  • The discharging process also follows an exponential decay equation.

Slide 30: Applications of Capacitors

  • Capacitors have various applications in electrical and electronic devices.
  • They are used in power supplies, filters, oscillators, amplifiers, radios, televisions, computers, and many other electronic circuits.
  • Capacitors also play a crucial role in energy storage systems such as electric vehicles and renewable energy technologies.