Electric Field And Potential And Concept Of Capacitance - Capacitor & Capacitance

• Electric Field: The region around an electric charge where a force is experienced by other charges.
• Electric Field Intensity: The force experienced per unit positive charge placed at a point.
• Electric Potential: The amount of work done in bringing a unit positive charge from infinity to a point in the electric field.
• Equipotential Surface: A surface in an electric field where all points have the same electric potential.
• Electric Potential Difference: The difference in electric potential between two points in an electric field.

Capacitor and Capacitance

• Capacitance: The ability of a capacitor to store electrical energy in the form of an electric field.
• Capacitor: A passive two-terminal electronic component used to store energy in an electric field.
• Dielectric: An insulating material placed between the plates of a capacitor, which increases the capacitance.
• Parallel Plate Capacitor: A type of capacitor consisting of two conductive plates separated by a dielectric material.
• Cylindrical Capacitor: A type of capacitor consisting of a cylindrical conductor surrounded by another concentric cylindrical conductor or a cylindrical dielectric.

Capacitance of Parallel Plate Capacitor

• The capacitance of a parallel plate capacitor can be calculated using the formula:
  • C = ε₀A/d
    • C: Capacitance
    • ε₀: Permittivity of free space (8.854 × 10⁻¹² F/m)
    • A: Area of the plates
    • d: Distance between the plates
• The capacitance is directly proportional to the area of the plates and inversely proportional to the distance between the plates.
• The larger the area and smaller the distance, the larger the capacitance.

Effect of Dielectric on Capacitance

• Placing a dielectric between the plates of a parallel plate capacitor increases its capacitance.
• The capacitance of a capacitor with a dielectric can be calculated using the formula:
  • C’ = kC
    • C’: Capacitance with dielectric
    • k: Dielectric constant of the material
    • C: Capacitance without dielectric
• The dielectric constant is a measure of how easily a material can be polarized by the electric field.

Energy Stored in a Capacitor

• The energy stored in a capacitor can be calculated using the formula:
  • U = ½CV²
    • U: Energy stored
    • C: Capacitance
    • V: Voltage across the capacitor
• The energy stored in the capacitor is directly proportional to the capacitance and the square of the voltage.
• The larger the capacitance or the voltage, the more energy can be stored in the capacitor.

Combination of Capacitors

• Capacitors can be connected in series or in parallel to form combinations.
• When capacitors are connected in series, the overall capacitance decreases.
• The equivalent capacitance of capacitors connected in series can be calculated using the formula:
  • 1/Ceq = 1/C₁ + 1/C₂ + 1/C₃ + …
    • Ceq: Equivalent capacitance
    • C₁, C₂, C₃, …: Capacitances of individual capacitors

Combination of Capacitors (Contd.)

• When capacitors are connected in parallel, the overall capacitance increases.
• The equivalent capacitance of capacitors connected in parallel can be calculated using the formula:
  • Ceq = C₁ + C₂ + C₃ + …
    • Ceq: Equivalent capacitance
    • C₁, C₂, C₃, …: Capacitances of individual capacitors
• Combinations of capacitors can be used to achieve desired capacitance values in practical circuits.

Charging and Discharging of a Capacitor

• When a capacitor is connected to a voltage source, it charges up as it accumulates charge.
• The time taken for a capacitor to charge to a certain percentage of its final charge depends on the time constant.
• The time constant (τ) can be calculated using the formula:
  • τ = RC
    • τ: Time constant
    • R: Resistance
    • C: Capacitance
• When a charged capacitor is disconnected from a voltage source, it discharges over time.
• The time taken for a capacitor to discharge to a certain percentage of its initial charge also depends on the time constant.

Summary

• Electric fields surround electric charges and create electric potentials.
• Capacitors store electrical energy in the form of an electric field.
• The capacitance of a capacitor depends on its physical properties and the presence of a dielectric.
• Capacitors can be connected in series or parallel to achieve desired capacitance values.
• Charging and discharging of capacitors follow exponential decay curves determined by the time constant.

11. Capacitors in Series

• When capacitors are connected in series, the total capacitance decreases.
• The voltage across each capacitor is the same.
• The reciprocal of the equivalent capacitance is equal to the sum of the reciprocals of the individual capacitances.

12. Capacitors in Series (Contd.)

• Formula for the equivalent capacitance of capacitors connected in series:
  • 1/Ceq = 1/C₁ + 1/C₂ + 1/C₃ + … Example:
• Capacitor C₁ = 4 μF
• Capacitor C₂ = 6 μF
• Capacitor C₃ = 8 μF

13. Capacitors in Series (Contd.)

• Applying the formula for capacitors in series:
  • 1/Ceq = 1/4μF + 1/6μF + 1/8μF
• Calculating the equivalent capacitance:
  • 1/Ceq = 3/12μF + 2/12μF + 1/12μF
  • 1/Ceq = 6/12μF
  • Ceq = 12/6μF
  • Ceq = 2μF

14. Capacitors in Parallel

• When capacitors are connected in parallel, the total capacitance increases.
• The voltage across each capacitor is the same.
• The sum of the individual capacitances is equal to the equivalent capacitance.

15. Capacitors in Parallel (Contd.)

• Formula for the equivalent capacitance of capacitors connected in parallel:
  • Ceq = C₁ + C₂ + C₃ + … Example:
• Capacitor C₁ = 5 μF
• Capacitor C₂ = 7 μF
• Capacitor C₃ = 3 μF

16. Capacitors in Parallel (Contd.)

• Applying the formula for capacitors in parallel:
  • Ceq = 5μF + 7μF + 3μF
  • Ceq = 15μF

17. Charging of a Capacitor

• When a capacitor is connected to a voltage source, it charges up and stores electrical energy.
• The charging process follows an exponential curve.
• The time it takes for a capacitor to charge to a certain percentage of its final charge is determined by the time constant.

18. Time Constant for Charging

• Formula for the time constant (τ) for charging a capacitor:
  • τ = RC
    • R: Resistance connected in series with the capacitor
    • C: Capacitance of the capacitor Example:
• Resistance R = 10 kΩ
• Capacitance C = 20 μF

19. Time Constant (Contd.)

• Applying the formula for the time constant:
  • τ = (10 kΩ)(20 μF)
  • τ = 200 ms

20. Discharging of a Capacitor

• When a charged capacitor is disconnected from a voltage source, it discharges over time.
• The discharging process also follows an exponential curve.
• The time it takes for a capacitor to discharge to a certain percentage of its initial charge is determined by the time constant.

21. Energy Stored in a Capacitor

• The energy stored in a capacitor can be calculated using the formula:
  • U = ½CV²
    • U: Energy stored
    • C: Capacitance
    • V: Voltage across the capacitor
• Example:
  • Capacitance C = 10 μF
  • Voltage V = 100 V
• Calculating the energy stored:
  • U = ½(10 μF)(100 V)²
  • U = ½(10 × 10⁻⁶ F)(100)²
  • U = ½(10⁻⁴ C)(10,000 V)
  • U = 5 × 10⁻² J

22. Combination of Capacitors

• Capacitors can be connected in series or in parallel to form combinations.
• When capacitors are connected in series, the overall capacitance decreases.
• The equivalent capacitance of capacitors connected in series can be calculated using the formula:
  • 1/Ceq = 1/C₁ + 1/C₂ + 1/C₃ + …

23. Combination of Capacitors (Contd.)

• Example:
  • Capacitor C₁ = 2 μF
  • Capacitor C₂ = 4 μF
  • Capacitor C₃ = 6 μF
• Calculating the equivalent capacitance:
  • 1/Ceq = 1/2μF + 1/4μF + 1/6μF
  • 1/Ceq = 3/6μF + 2/6μF + 1/6μF
  • 1/Ceq = 6/6μF
  • Ceq = 6/6μF
  • Ceq = 1μF

24. Combination of Capacitors (Contd.)

• When capacitors are connected in parallel, the overall capacitance increases.
• The equivalent capacitance of capacitors connected in parallel can be calculated using the formula:
  • Ceq = C₁ + C₂ + C₃ + …

25. Combination of Capacitors (Contd.)

• Example:
  • Capacitor C₁ = 3 μF
  • Capacitor C₂ = 5 μF
  • Capacitor C₃ = 7 μF
• Calculating the equivalent capacitance:
  • Ceq = 3μF + 5μF + 7μF
  • Ceq = 15μF

26. Charging and Discharging of a Capacitor

• When a capacitor is connected to a voltage source, it charges up as it accumulates charge.
• The time taken for a capacitor to charge to a certain percentage of its final charge depends on the time constant.
• The time constant (τ) can be calculated using the formula:
  • τ = RC
    • R: Resistance
    • C: Capacitance

27. Charging and Discharging of a Capacitor (Contd.)

• When a charged capacitor is disconnected from a voltage source, it discharges over time.
• The time taken for a capacitor to discharge to a certain percentage of its initial charge also depends on the time constant.
• Example:
  • Resistance R = 5 kΩ
  • Capacitance C = 10 μF

28. Charging and Discharging of a Capacitor (Contd.)

• Calculating the time constant for charging and discharging:
  • τ = (5 kΩ)(10 μF)
  • τ = 50 ms

29. Summary

• Electric fields surround electric charges and create electric potentials.
• Capacitors store electrical energy in the form of an electric field.
• The capacitance of a capacitor depends on its physical properties and the presence of a dielectric.
• Capacitors can be connected in series or parallel to achieve desired capacitance values.
• Charging and discharging of capacitors follow exponential decay curves determined by the time constant.

30. Additional Resources

• Websites:

  • Khan Academy
  • Physics Classroom
  • HyperPhysics

• Books:

  • “Fundamentals of Physics” by David Halliday, Robert Resnick, and Jearl Walker
  • “University Physics” by Hugh D. Young and Roger A. Freedman