Slide 1

  • Topic: Cylindrical and Spherical Capacitors, Series and Parallel Combinations
  • Subtopic: Parallel Combination - Formula derivation

Slide 2

  • Cylindrical and Spherical Capacitors are widely used in electrical circuits.
  • They are capable of storing and releasing electrical energy.
  • Capacitance is the ability of a capacitor to store electric charge.
  • Capacitance is defined as the ratio of charge stored on each plate to the potential difference applied across the plates.

Slide 3

  • Cylindrical capacitors consist of two concentric cylindrical conductors.
  • The inner cylinder acts as the positive plate, and the outer cylinder acts as the negative plate.
  • The charge is stored on the inner cylinder, and the electric field is established between the two cylinders.
  • The capacitance of a cylindrical capacitor is given by the formula: C = 2πε₀L / ln(b/a), where C is the capacitance, ε₀ is the permittivity of free space, L is the length of the capacitor, a is the radius of the inner cylinder, and b is the radius of the outer cylinder.

Slide 4

  • Spherical capacitors consist of two concentric spherical conductors.
  • The inner sphere acts as the positive plate, and the outer sphere acts as the negative plate.
  • The charge is stored on the inner sphere, and the electric field is established between the two spheres.
  • The capacitance of a spherical capacitor is given by the formula: 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 5

  • Series combination of capacitors occurs when the positive plate of one capacitor is connected to the negative plate of another capacitor.
  • The charge on each capacitor is the same, given as Q.
  • The potential difference across the entire combination is the sum of the potential differences across each capacitor, given as V = V₁ + V₂ + V₃ + … + Vn.

Slide 6

  • The total capacitance in a series combination can be calculated using the formula: 1/C = 1/C₁ + 1/C₂ + 1/C₃ + … + 1/Cn, where C is the total capacitance, C₁, C₂, C₃, … , Cn are the individual capacitances.

Slide 7

  • Parallel combination of capacitors occurs when the positive plates of all capacitors are connected together, and the negative plates are connected together.
  • The potential difference across each capacitor is the same, given as V.
  • The total charge stored in the combination is the sum of the charges on each capacitor, given as Q = Q₁ + Q₂ + Q₃ + … + Qn.

Slide 8

  • The total capacitance in a parallel combination can be calculated using the formula: C = C₁ + C₂ + C₃ + … + Cn, where C is the total capacitance, C₁, C₂, C₃, … , Cn are the individual capacitances.

Slide 9

  • When capacitors are connected in parallel, the equivalent capacitance increases.
  • This is due to the increased ability of the combination to store electric charge.
  • The total capacitance is the sum of the individual capacitances.
  • Capacitors in parallel have the same potential difference across them.

Slide 10

  • In a parallel combination of capacitors, the larger the individual capacitance of a capacitor, the more charge it can store for a given potential difference.
  • Parallel combination of capacitors is used to increase the overall capacitance in electronic circuits.
  • It is commonly used in power supply units and energy storage devices.
  • The energy stored in a capacitor is given by the formula: E = 1/2CV², where E is the energy, C is the capacitance, and V is the potential difference.

Slide 11

  • Parallel Combination Formula derivation:

    • Let C₁, C₂, C₃, …, Cn be the individual capacitances of the capacitors connected in parallel.

    • The total charge Q stored in the parallel combination is the sum of the charges on each capacitor, Q = Q₁ + Q₂ + Q₃ + … + Qn.

    • Using the equation Q = CV, where C is the capacitance and V is the potential difference,

    • Q₁ = C₁V, Q₂ = C₂V, Q₃ = C₃V, …, Qn = CnV.

    • Substituting the values of charges in the total charge equation, we get Q = C₁V + C₂V + C₃V + … + CnV.

    • Factoring out V, we get Q = V(C₁ + C₂ + C₃ + … + Cn).

    • The total capacitance C of the parallel combination is given by C = (C₁ + C₂ + C₃ + … + Cn).

    • Therefore, C = Q/V.

    • Combining the equations, we get C = (C₁ + C₂ + C₃ + … + Cn) = C₁ + C₂ + C₃ + … + Cn.

    • Hence, the total capacitance C of capacitors connected in parallel is equal to the sum of the individual capacitances C₁, C₂, C₃, …, Cn.

Slide 12

  • In a parallel combination of capacitors, the total capacitance C is always greater than the capacitance of any individual capacitor (C > C₁, C > C₂, C > C₃, …, C > Cn).
  • The equivalent capacitance in a parallel combination is always larger than the largest individual capacitance.
  • The larger the individual capacitances, the greater the overall capacitance.
  • Capacitors in parallel have the same potential difference across them.
  • The inverse of the total capacitance is equal to the sum of the inverses of the individual capacitances: 1/C = 1/C₁ + 1/C₂ + 1/C₃ + … + 1/Cn.

Slide 13

  • Example:
    • Consider two capacitors with capacitances C₁ = 4μF and C₂ = 6μF connected in parallel.
    • The total capacitance C of the parallel combination is given by C = C₁ + C₂ = 4μF + 6μF = 10μF.
    • Therefore, the total capacitance of the combination is 10μF.
  • The parallel combination of capacitors is frequently used in electronic circuits to increase the overall capacitance and store more charge for a given potential difference.

Slide 14

  • Capacitors connected in parallel have the same potential difference across them.
  • The energy stored in a capacitor can be calculated using the formula: E = 1/2CV², where E is the energy, C is the capacitance, and V is the potential difference.
  • In a parallel combination, the potential difference is the same across all the capacitors.
  • Therefore, the energy stored in each capacitor is given by E = 1/2C₁V², E = 1/2C₂V², E = 1/2C₃V², …, E = 1/2CnV².

Slide 15

  • The total energy stored in the parallel combination is obtained by summing the energies of each individual capacitor, E = E₁ + E₂ + E₃ + … + En.
  • Substituting the values of energies, we get E = 1/2C₁V² + 1/2C₂V² + 1/2C₃V² + … + 1/2CnV².
  • Factoring out V², we get E = 1/2V²(C₁ + C₂ + C₃ + … + Cn).
  • The total capacitance C of the parallel combination is given by C = C₁ + C₂ + C₃ + … + Cn.
  • Therefore, E = 1/2C(V²).
  • Hence, the total energy stored in capacitors connected in parallel is equal to 1/2CV², where C is the total capacitance and V is the potential difference.

Slide 16

  • Advantages of using capacitors in parallel combination:
    • Increased capacitance: The total capacitance of the parallel combination is the sum of the individual capacitances, resulting in a larger overall capacitance.
    • Greater charge storage: With a higher capacitance, the capacitors can store more charge for a given potential difference.
    • Enhanced energy storage: The total energy stored in a parallel combination is higher due to the increased capacitance.

Slide 17

  • Applications of parallel combination of capacitors:
    • Power supply units: Capacitors connected in parallel are used to smooth out voltage fluctuations and provide a stable DC output.
    • Energy storage devices: Parallel combination of capacitors increases the overall capacitance, which is useful in devices like energy storage systems and backup power supplies.
    • Audio systems: Parallel capacitors are used to tune audio systems by adjusting the overall capacitance and improving the quality of sound reproduction.

Slide 18

  • Parallel combination of capacitors is also known as “capacitor banks” in larger electrical systems.
  • These capacitor banks are utilized for power factor correction, voltage regulation, and improving the efficiency of the electrical grid.
  • Advanced industrial applications also use parallel combinations of capacitors to compensate for reactive power and reduce losses.

Slide 19

  • Disadvantages of parallel combination of capacitors:
    • Larger physical size: As the number of capacitors connected in parallel increases, the physical size of the combination may become larger.
    • Cost: Using multiple capacitors adds to the cost of the circuit or system.
    • Complex wiring: More capacitors require additional wiring connections, increasing the complexity of the circuit.

Slide 20

  • Recap:
    • In a parallel combination of capacitors, the total capacitance is the sum of the individual capacitances.
    • The larger the individual capacitances, the greater the overall capacitance.
    • Capacitors in parallel have the same potential difference across them.
    • The inverse of the total capacitance is equal to the sum of the inverses of the individual capacitances.
    • The energy stored in a capacitor is given by the formula E = 1/2CV², where E is the energy, C is the capacitance, and V is the potential difference.

Slide 21

  • Parallel Combination: Formula derivation (continued)
    • Let C₁, C₂, C₃, …, Cn be the individual capacitances of the capacitors connected in parallel.
    • The total capacitance C of the parallel combination is given by C = C₁ + C₂ + C₃ + … + Cn.
    • To derive this formula, let’s consider two capacitors connected in parallel.
    • Capacitor 1 has a charge Q₁ and a potential difference V across it.
    • Capacitor 2 has a charge Q₂ and the same potential difference V across it.
    • The total charge on the combination is the sum of the charges on each capacitor, Q = Q₁ + Q₂.
    • The total capacitance C of the combination is given by C = Q/V.
    • Substituting the values of charges and potential difference, we get C = (Q₁ + Q₂)/V.
    • Using the equation Q = CV, we can rewrite it as C = (C₁ + C₂)V/V.
    • Finally, cancelling out the common factor V, we get C = C₁ + C₂.

Slide 22

  • In a parallel combination of capacitors, the total capacitance C is the sum of the individual capacitances C₁, C₂, C₃, …, Cn.
  • The unit of capacitance is Farad (F).
  • The parallel combination of capacitors is frequently used to increase the overall capacitance and store more charge for a given potential difference.
  • It is essential to select capacitors with the same voltage rating when connecting them in parallel.
  • When capacitors are connected in parallel, the positive terminal of each capacitor should be connected to the positive terminal of the battery, and the negative terminals should be connected together in a similar manner.

Slide 23

  • Example:
    • Consider three capacitors with capacitances C₁ = 2μF, C₂ = 4μF, and C₃ = 6μF connected in parallel.
    • The total capacitance C of the parallel combination is given by C = C₁ + C₂ + C₃ = 2μF + 4μF + 6μF = 12μF.
    • Therefore, the total capacitance of the combination is 12μF.
  • Capacitors connected in parallel are commonly used in filter circuits, audio systems, electronic devices, and power supply units.

Slide 24

  • Parallel combination of capacitors in circuit diagrams is represented as follows: ________ | | C₁ C₂ | | --
  • The parallel combination symbol consists of two parallel lines with capacitors connected to each line.

Slide 25

  • Potential difference across capacitors in parallel combination:
    • In a parallel combination, each capacitor has the same potential difference across it, which is equal to the potential difference across the combination.
    • The potential difference across the combination remains the same as the potential difference provided by the battery or the power supply.
    • The voltage rating of the capacitors should always be greater than or equal to the potential difference across the combination to avoid damage.

Slide 26

  • Difference between series and parallel combination of capacitors: Series Combination:

    • The positive plate of one capacitor is connected to the negative plate of another capacitor.
    • The total capacitance decreases as the inverse of the sum of the inverses of individual capacitances.
    • The charge on each capacitor is the same, but the potential difference across each capacitor is different.

    Parallel Combination:

    • The positive plates of all capacitors are connected together, and the negative plates are connected together.
    • The total capacitance increases as the sum of the individual capacitances.
    • The potential difference across each capacitor is the same, but the charge on each capacitor is different.

Slide 27

  • Capacitors in parallel can be replaced by a single equivalent capacitor.
  • The equivalent capacitance is equal to the sum of the individual capacitances.
  • The equivalent capacitance allows easier analysis of complex circuits and simplifies calculations.
  • Adding capacitors in parallel increases their ability to store charge and energy.

Slide 28

  • Precautions when working with capacitors in parallel:
    • Ensure all capacitors have the same voltage rating as the potential difference across the combination.
    • Verify that the capacitors are connected with the correct polarity.
    • Avoid touching the leads of the capacitors while they are charged to prevent electrical shock.
    • In high-voltage applications, use appropriate safety measures and precautions.

Slide 29

  • Recap:
    • Capacitors connected in parallel have the same potential difference across them.
    • The total capacitance in a parallel combination is equal to the sum of the individual capacitances.
    • When capacitors are connected in parallel, their total capacitance increases.
    • Parallel combination of capacitors is used to increase the overall capacity of energy storage and smoothen out voltage fluctuations in electrical systems.
    • Equivalent capacitance can be used to simplify complex circuits and calculations.

Slide 30

  • Thank you for attending this lecture on the parallel combination of capacitors.
  • By understanding the concepts presented and practicing problem-solving, you can master this topic.
  • Be sure to review the formulas, work through examples, and ask questions when needed.
  • Good luck with your studies and preparations for the 12th Boards exam!