Capacitance

Capacitance

• a) Parallel Plate Capacitor: Basic Concept

• A parallel plate capacitor is a simple electrical component that consists of two equal, parallel metal plates separated by a small distance. The space between these plates can be filled with:
• 1. Vacuum (ideal case)
• 2. Air (practical case)
• 3. A dielectric material (to enhance performance)
• ⇒ Structure of a Parallel Plate Capacitor
• The two metal plates are conductive (e.g., aluminum, copper).
• The plates are placed parallel to each other, ensuring uniform charge distribution.
• The gap between the plates is filled with a medium (vacuum, air, or dielectric).
• Terminals are connected to a voltage source, allowing charge storage.
• 
• Figure 1 Parallel plate capacitor
• ⇒ Works
• When a voltage is applied, one plate becomes positively charged (+Q) while the other becomes negatively charged (-Q).
• This creates an electric field between the plates.
• The capacitor stores electrical energy by maintaining this charge separation.
• This simple setup forms the basis of capacitors used in various electronic applications, such as energy storage, signal processing, and circuit stabilization.

• b) A Capacitor Stores Energy by Transferring Charge

• When a voltage source (such as a battery) is connected to the capacitor:
• 1. Electrons move due to the applied potential difference.
• 2. The plate connected to the positive terminal of the battery loses electrons and becomes positively charged (+Q).
• 3. The plate connected to the negative terminal of the battery gains electrons and becomes negatively charged (−Q).
• 4. This creates an electric field (E) between the plates, which stores energy.
• 
• Figure 2  Capacitor store energy by transferring charge
• Since the charge on both plates is equal and opposite, the total charge in the capacitor is zero, but an electric potential difference is maintained, allowing the capacitor to store energy.
• The energy stored in a capacitor is given by:
• [math]U = \frac{1}{2} C V^2[/math]
• Where:
• – U = stored energy (joules)
• – C = capacitance (farads)
• – V = voltage across the plates (volts)

• c) Definition of Capacitance:[math]C = \frac{Q}{V}[/math]

• The capacitance (C) of a capacitor is a measure of its ability to store charge per unit voltage. It is defined as:
• [math]C = \frac{Q}{V}[/math]
• Where:
• – C = capacitance (farads, F)
• – Q = charge stored on one plate (coulombs, C)
• – V = voltage across the plates (volts, V)
• A higher capacitance means more charge can be stored for a given voltage.
• A larger applied voltage results in greater charge storage, but the capacitance remains constant for a given capacitor.
• The stored energy U in terms of the capacitance by substituting Q = CV into the above equation. This given two more equations for U:
• [math]U = \frac{QV}{2} \\  U = \frac{(C V) V}{2} \\  U = \frac{C V^2}{2} \\  U = \frac{Q^2}{2C}[/math]

• d) Formula for Capacitance of a Parallel Plate Capacitor with No Dielectric

• For a parallel plate capacitor with plates separated by vacuum or air, the capacitance is given by:
• [math]C = \frac{\varepsilon_0 A}{d}[/math]
• Where:
• – C = capacitance (farads, F)
• – [math]\varepsilon_0[/math] = permittivity of free space ([math]8.85 \times 10^{-12} \text{ F/m}[/math])
• – A = area of each plate (m²)
• – d = separation between the plates (m)
• ⇒ Understanding the Formula:
• Larger Plate Area (A) → Higher Capacitance (more charge storage).
• Smaller Plate Separation (d) → Higher Capacitance (stronger electric field).
• Air or Vacuum as Medium: No additional dielectric effect.
• ⇒ Practical Implications:
• Increasing plate size or reducing separation makes the capacitor store more charge.
• If a dielectric material is inserted, the capacitance increases, as the material increases the permittivity.

• e) A Dielectric Increases the Capacitance of a Parallel Plate Capacitor

• A dielectric is an insulating material placed between the plates of a capacitor to increase its capacitance. In a vacuum-spaced capacitor, the capacitance is given by:
• [math]C_0 = \frac{\varepsilon_0 A}{d}[/math]
• Where:
• – [math]C_0[/math] = capacitance without a dielectric (vacuum or air)
• – [math]\varepsilon_0[/math]​ = permittivity of free space ([math]8.85 \times 10^{-12} \text{ F/m}[/math])
• – A = plate area (m²)
• – d = plate separation (m)
• ⇒ Effect of a Dielectric
• When a dielectric material (such as plastic, ceramic, or glass) is ins>When a dielectric material (such as plastic, ceramic, or glass) is inserted betweerted between the plates, the new capacitance becomes:
• [math]C = \varepsilon_r C_0 \\
  C = \frac{\varepsilon_r \varepsilon_0 A}{d}[/math]
• Where:
• – [math]\varepsilon_r[/math]= relative permittivity (dielectric constant) of the material (​[math]\varepsilon_r[/math]>1)
• – C = new capacitance with the dielectric
• 
• Figure 3  Parallel plate capacitors
• When a dielectric material (such as plastic, ceramic, or glass) is inserted betwe>Work = Force x distance, the work done on the particle is W = qEd using the fact that distance between plates is d. the potential difference between two points is defined as the change in potential energy per unit charge, so dividing W by q gives
• V = Ed
• The charges accumulate on each plate, the increasing electric field causes an increased potential difference between the plates. “The plates will continue to charge until the potential difference across them is equal to the potential difference supplied by the cell”.
• ⇒ Capacitance Increase:
• 1. Dielectric Polarization:
• – The dielectric molecules align with the electric field, reducing the effective field strength.
• 2. Reduces Voltage (V) for Same Charge (Q):
• – Since the dielectric opposes the field, it requires less voltage to store the same charge.
• 3. Stores More Charge at Same Voltage:
• – If voltage is constant, the charge stored increases, increasing capacitance.
• Adding a dielectric increases capacitance by a factor of [math][/math], allowing capacitors to store more charge and energy.

• f) Uniform Electric Field in a Parallel Plate Capacitor

• The electric field (E) in a parallel plate capacitor is uniform between the plates, meaning:
• – It has a constant magnitude and direction.
• – The field lines are parallel and evenly spaced.
• ⇒ Equation for the Electric Field
• For a parallel plate capacitor, the electric field is given by:
• [math]E = \frac{V}{d}[/math]
• Where:
• – E = electric field strength (V/m)
• – V = voltage across the plates (volts)
• – d = plate separation (m)
• ⇒ Understanding the Formula
• – Larger voltage (V) → Stronger electric field.
• – Greater plate separation (d) → Weaker electric field.
• The field is strongest near the plates and remains constant between them (assuming a uniform dielectric or vacuum).

• g) Energy Stored in a Capacitor

• A capacitor stores electrical energy in its electric field. The total energy stored in a capacitor is given by:
• [math]U = \frac{1}{2}QV[/math]
• Where:
• – U = energy stored (joules)
• – Q = charge stored on the capacitor (coulombs)
• – V = voltage across the plates (volts)
• Alternative Forms of the Energy Equation
• Using Q = CV, we can express energy in different forms:
• 1. In terms of capacitance and voltage:
• [math]U = \frac{1}{2}QV^2[/math]
• – Shows that increasing capacitance or voltage increases stored energy.
• 2. In terms of capacitance and charge:
• [math]U = \frac{Q^2}{2C}[/math]
• – Highlights that for a given charge, a higher capacitance results in less stored energy, as the capacitor distributes the charge more efficiently.

• h) Equations for Capacitors in Series and Parallel

• Capacitors can be arranged in series or parallel, and their total capacitance follows specific formulas.
• ⇒ Capacitors in Parallel
• When capacitors are connected in parallel, the total capacitance is the sum of individual capacitances:
• [math]C_{\text{eq}} = C_1 + C_2 + C_3 + C_4 + \dots[/math]
• Where:
• – [math]C_{\text{eq}}[/math]​ = equivalent capacitance of the parallel combination
• – [math]C_1 , C_2 , C_3 , C_4[/math] are the individual capacitances
• The voltage across each capacitor is the same.
• Each capacitor stores charge according to Q = CV
• [math]Q_{\text{tot}} = C_1 V + C_2 V + C_3 V + \dots \\
  \frac{Q_{\text{tot}}}{V} = C_1 + C_2 + C_3 + \dots[/math]
• The total charge stored is the sum of individual charges.
• 
• Figure 4 Parallel plate capacitor
• – Parallel capacitors increase total capacitance.
• – Used in power supplies and filter circuits for stable voltage.
• ⇒ Capacitors in Series
• When capacitors are connected in series, the total capacitance is given by:
• [math]\frac{1}{C_{\text{eq}}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + \dots[/math]
• Where:
• – [math]C_{\text{eq}}[/math]​ = equivalent capacitance of the series combination
• – [math]C_1 , C_2 , C_3 , \dots[/math]are the individual capacitances
• The charge on each capacitor is the same.
• The voltage divides among capacitors.
• The total capacitance is always less than the smallest capacitor.
• 
• Figure 5 Series of capacitor
• Series capacitors decrease total capacitance.
• Used in high-voltage circuits to handle large potentials.
• The potential difference across each capacitor is Vi where i represents the i’th capacitor, then
• [math]V = V_1 + V_2 + V_3 [/math]
• The capacitor must all hold the same charge seeing as the charge lost from one plate must have two gained by the other in the charging process. Diving by this charge gives
• [math]\frac{V}{Q} = \frac{V_1}{Q} + \frac{V_2}{Q} + \frac{V_3}{Q}[/math]

• i) Charging and Discharging of a Capacitor Through a Resistor

• A capacitor charges and discharges through a resistor in an RC circuit, where:
• – R = resistance in ohms (Ω)
• – C = capacitance in farads (F)
• – RC = time constant (seconds), which determines how fast the capacitor charges/discharges
• ⇒ Charging Process
• When a capacitor charges through a resistor, it starts with no charge and gradually accumulates charge until it reaches its maximum value.
• Initially (t=0): No charge on the capacitor, voltage is zero.
• As time increases: Charge builds up, and the voltage across the capacitor increases.
• At t→∞: The capacitor is fully charged and acts as an open circuit.
• The charge on the capacitor at time t is given by:
• [math]Q = Q_0 \left( 1 – e^{-t / RC} \right)[/math]
• Where:
• – [math]Q_0[/math] = maximum charge stored ([math]Q_0​=CV[/math])
• – t = time (s)
• – e = Euler’s number (≈2.718)
• – RC = time constant
• 
• Figure 6 Charging of the capacitor
• ⇒ Understanding the Charging Equation
• When t=0, Q=0 (initially uncharged)
• As [math]t \to \infty, \quad Q \to Q_0[/math]​ (fully charged capacitor).
• The time constant RC determines the rate of charging.
• After , the capacitor is about 63% charged.
   After 5RC, it is over 99% charged.
• ⇒ Discharging Process
• When the voltage source is removed, the capacitor discharges through the resistor, releasing stored energy.
• 1. Initially (t=0): The capacitor is fully charged.
• 2. As time increases: The charge decreases as current flows through the resistor.
• 3. At t→∞: The capacitor is fully discharged, and voltage drops to zero.
• The charge on the capacitor during discharge is given by:
• [math]Q = Q_0 e^{-t / RC}[/math]
• Where:
• – ​[math]Q_0[/math] = initial charge
• – t = time
• – RC = time constant
• 
• Figure 7 Discharging of the capacitor
• ⇒ Understanding the Discharging Equation
• At t=0,  ​ (fully charged).
• As t→∞, Q=0 (fully discharged).
• The charge decreases exponentially.
• Large time constants cause slower discharges. Seeing as the potential difference and current are both proportional to the charge left on the capacitor, both quantities decay in a similar way:
• [math]V = V_0 e^{-t / RC} \\
  I = I_0 e^{-t / RC}[/math]
• After t = RC, the capacitor retains about 37% of its charge.
   After , it is over 99% discharged.

• j) The Time Constant RC

• The time constant is a key factor in capacitor circuits:
• ​[math]\tau = RC[/math]
• Where:
• – τ = time constant (seconds)
• – R = resistance (Ω)
• – C = capacitance (F)
• 
• Figure 8 Time constant RC circuit
• ⇒ Significance of RC:
• Determines the speed of charging and discharging.
• Larger RC → Slower process (longer time to charge/discharge).
• Smaller RC → Faster process (shorter time to charge/discharge).

• Specified Practical Work

• 1. Investigation of the Charging and Discharging of a Capacitor to Determine the Time Constant RC

• ⇒ Objective:
• To experimentally determine the time constant RC of a capacitor by analyzing its charging and discharging behavior through a resistor.
• ⇒ Apparatus Required:
• – A capacitor (C)
• – A resistor (R)
• – A DC power supply (battery or voltage source)
• – A switch (to control charging and discharging)
• – A voltmeter (or an oscilloscope)
• – A stopwatch (or data-logging device)
• – Connecting wires
• 
• Figure 9 Charging and discharging of capacitor
• ⇒ Experimental Procedure:
• 1. Charging the Capacitor:
• 1. Set up the circuit as shown below:
• – Connect the capacitor C in series with the resistor R and a DC voltage source V.
• – Attach a voltmeter across the capacitor.
• – Include a switch to control charging and discharging.
• 2. Close the switch to start charging the capacitor.
• 3. Record the voltage [math]V_C[/math]​ across the capacitor at regular time intervals (t).
• 4. Plot [math]V_C[/math]  vs. t and compare it with the theoretical equation:
• [math]V_C = V_0 \left( 1 – e^{-t / RC} \right)[/math]
• 5. Determine the time constant RC from the graph:
• – The voltage reaches 63% of the supply voltage after one–time constant t = RC.
• 2. Discharging the Capacitor:
• 1. Open the switch to disconnect the voltage source and allow the capacitor to discharge through the resistor.
• 2. Record the voltage across the capacitor at regular time intervals.
• 3. Plot [math]V_C[/math]  vs. t and compare with the equation:
• [math]V_C = V_0 e^{-t / RC}[/math]
• 4. Identify RC from the graph:
• The voltage drops to 37% of the initial voltage after one–time constant t = RC
• ⇒ Analysis and Conclusion:
• By measuring how fast the voltage changes, we can determine the experimental time constant.
• Compare the experimental RC with the calculated value using
• [math]RC = \frac{1}{\text{slope of } \ln(V) \text{ vs } t \text{ graph}}[/math]
• Sources of error: Resistance of wires, capacitor leakage, or imperfect connections.
• 2. Investigation of the Energy Stored in a Capacitor
• ⇒ Objective:
• To measure the energy stored in a capacitor and compare it with the theoretical equation:
• [math]U = \frac{1}{2} C V^2[/math]
• ⇒ Apparatus Required:
• – A capacitor (C)
• – A DC power supply (V)
• – A resistor (R)
• – A switch
• – A joule meter or voltmeter & ammeter
• – A stopwatch
• 
• Figure 10 Energy store in a capacitor
• ⇒ Experimental Procedure:
• 1. Charging the Capacitor:
• i. Set up the circuit as before, with a capacitor, resistor, and power source.
• ii. Fully charge the capacitor by closing the switch and applying voltage V.
• iii. Measure the voltage V across the capacitor once it is fully charged.
• 2. Discharging the Capacitor into a Load (Energy Transfer Test):
• i. Disconnect the power supply and connect a small resistor across the capacitor to let it discharge.
• ii. Measure the current (I) and voltage (V) across the resistor as the capacitor discharges.
• iii. Calculate the energy dissipated in the resistor using:
• [math]E = \int P \, dt \\ E = \int V I \, dt[/math]
• (Where [math]P = VI[/math] is the power dissipated in the resistor).
• iv. Compare with the theoretical energy stored in the capacitor:
• [math]U = \frac{1}{2} C V^2[/math]
• ⇒ Analysis and Conclusion:
• The energy measured should be close to the theoretical value.
• – Some energy loss occurs due to resistance and heat dissipation.
• – The capacitor’s ability to store energy efficiently can be verified.
• 3. Summary of Experimental Results

• These experiments help in understanding capacitor behavior, time constants, and energy storage in practical circuits