Lab Manual: Verification of Laws of Capacitances

VERIFICATION OF LAWS OF CAPACITANCES BY REPEATED CHARGE DECAY METHOD

1. AIM

To verify the laws of capacitance (series and parallel combinations) using the repeated charge decay method and to determine the effective capacitance of various combinations.

2. APPARATUS USED

  • Capacitors of different capacitance values (e.g., 1μF, 2μF, 4μF, etc.)
  • Ballistic galvanometer
  • High resistance (~1MΩ)
  • Battery (6V)
  • Discharge key
  • Charge key
  • Connecting wires
  • Digital multimeter (optional - for measuring actual capacitance values)
  • Stopwatch (for time measurements)

3. DIAGRAM

Circuit diagram for verification of laws of capacitances

Figure 1: Experimental setup for verification of laws of capacitances

4. THEORY

The laws of capacitance describe how capacitors behave when connected in series or parallel combinations. A capacitor is a device that stores electrical energy in an electric field. The capacity of a capacitor to store charge is measured in farads (F).

Parallel Combination: When capacitors are connected in parallel, the effective capacitance is the sum of individual capacitances.

$C_{parallel} = C_1 + C_2 + C_3 + ... + C_n$

Series Combination: When capacitors are connected in series, the reciprocal of the effective capacitance is the sum of the reciprocals of individual capacitances.

$\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + \frac{1}{C_3} + ... + \frac{1}{C_n}$

Repeated Charge Decay Method: This method involves repeatedly charging a capacitor or a combination of capacitors to a fixed potential and then allowing it to discharge through a high resistance while measuring the discharge current using a ballistic galvanometer.

When a capacitor of capacitance C is charged to a potential V and then discharged through a ballistic galvanometer, the first deflection (θ₁) is proportional to the charge Q:

$Q = CV = k\theta_1$

where k is the ballistic constant of the galvanometer.

If we allow the capacitor to discharge through a high resistance R for a time t and then discharge it again through the galvanometer, the second deflection (θ₂) will be related to the remaining charge on the capacitor:

$Q' = Q \cdot e^{-\frac{t}{RC}} = k\theta_2$

Taking the ratio of these two deflections:

$\frac{\theta_2}{\theta_1} = e^{-\frac{t}{RC}}$

This relationship allows us to determine the capacitance C, as R and t are known, and the ratio of deflections can be measured. By applying this method to different capacitor combinations, we can verify the laws of capacitance.

5. FORMULA

For a single capacitor:

$C = \frac{-t}{R \cdot \ln(\frac{\theta_2}{\theta_1})}$

where:

  • C = Capacitance in farads
  • t = Time interval between successive discharges in seconds
  • R = High resistance in ohms
  • θ₁ = First deflection of galvanometer
  • θ₂ = Second deflection of galvanometer

For parallel combination:

$C_{parallel} = C_1 + C_2 + ... + C_n$

For series combination:

$C_{series} = \frac{C_1 \cdot C_2}{C_1 + C_2}$ (for two capacitors)

$\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$ (for multiple capacitors)

6. PROCEDURE

  1. Set up the circuit as shown in the diagram with a single capacitor C₁.
  2. Ensure the discharge key K₂ is open and close the charge key K₁ to charge the capacitor to the battery potential.
  3. Open the charge key K₁ and immediately close the discharge key K₂ to discharge the capacitor through the ballistic galvanometer.
  4. Record the first deflection (θ₁) of the galvanometer.
  5. Charge the capacitor again by closing K₁ and opening K₂.
  6. After charging, open K₁ and wait for a measured time t (e.g., 30 seconds).
  7. After time t, close K₂ to discharge the capacitor through the galvanometer and record the second deflection (θ₂).
  8. Repeat steps 5-7 for different time intervals t to obtain multiple readings.
  9. Calculate the capacitance C₁ using the formula provided.
  10. Repeat the above procedure for capacitor C₂.
  11. Connect capacitors C₁ and C₂ in parallel and repeat the procedure to find the effective capacitance C₁₂ₚ.
  12. Connect capacitors C₁ and C₂ in series and repeat the procedure to find the effective capacitance C₁₂ₛ.
  13. Calculate the theoretical values for parallel and series combinations.
  14. Compare the experimental values with the theoretical values and verify the laws of capacitance.

Note: For accurate results, ensure that the high resistance R is much greater than the galvanometer resistance, and the time intervals are chosen appropriately based on the RC time constant.

7. OBSERVATION TABLE

For Individual Capacitors:

Capacitor Time Interval (t) seconds First Deflection (θ₁) divisions Second Deflection (θ₂) divisions θ₂/θ₁ ln(θ₂/θ₁) Capacitance (μF)
C₁
C₂

For Capacitors in Parallel:

Combination Time Interval (t) seconds First Deflection (θ₁) divisions Second Deflection (θ₂) divisions θ₂/θ₁ ln(θ₂/θ₁) Capacitance (μF)
C₁ + C₂

For Capacitors in Series:

Combination Time Interval (t) seconds First Deflection (θ₁) divisions Second Deflection (θ₂) divisions θ₂/θ₁ ln(θ₂/θ₁) Capacitance (μF)
C₁ and C₂ in series

8. CALCULATIONS

Step 1: Calculate individual capacitances C₁ and C₂ using:

$C = \frac{-t}{R \cdot \ln(\frac{\theta_2}{\theta_1})}$

Step 2: Calculate the experimental capacitance for parallel and series combinations using the same formula.

Step 3: Calculate the theoretical capacitance for parallel combination:

$C_{parallel} = C_1 + C_2$

Step 4: Calculate the theoretical capacitance for series combination:

$C_{series} = \frac{C_1 \cdot C_2}{C_1 + C_2}$

Step 5: Calculate the percentage error:

$\text{Percentage Error} = \frac{|\text{Theoretical Value} - \text{Experimental Value}|}{\text{Theoretical Value}} \times 100\%$

Sample Calculation: For each case, show one complete sample calculation with the observed values substituted into the formulas.

Summary of Results:

Capacitor Combination Experimental Value (μF) Theoretical Value (μF) Percentage Error (%)
C₁ - -
C₂ - -
C₁ and C₂ in parallel
C₁ and C₂ in series

9. RESULT

  1. The capacitance of capacitor C₁ is ________ μF.
  2. The capacitance of capacitor C₂ is ________ μF.
  3. The effective capacitance of C₁ and C₂ in parallel is ________ μF (Experimental) and ________ μF (Theoretical).
  4. The effective capacitance of C₁ and C₂ in series is ________ μF (Experimental) and ________ μF (Theoretical).
  5. The percentage error for the parallel combination is ________%.
  6. The percentage error for the series combination is ________%.
  7. The laws of capacitance for series and parallel combinations have been verified with errors within acceptable limits.

10. PRECAUTIONS

  1. Before starting the experiment, discharge all capacitors completely to avoid any residual charge.
  2. Ensure all connections are tight and properly made.
  3. Use a high-quality, high-value resistor (R) with a known value.
  4. Ensure that the ballistic galvanometer is properly damped and calibrated.
  5. The time interval between successive discharges should be accurately measured.
  6. Maintain constant battery voltage throughout the experiment.
  7. Avoid touching the capacitor terminals during the experiment to prevent accidental discharge.
  8. Allow sufficient time for the galvanometer to return to zero between readings.
  9. Take multiple readings for each configuration and average the results for better accuracy.
  10. Ensure that the resistance of the galvanometer is much smaller than the high resistance (R) used in the circuit.
  11. The room temperature should be constant throughout the experiment as the capacitance may vary with temperature.
  12. Ensure that the capacitors used are in good condition and do not have significant leakage.

11. VIVA VOICE QUESTIONS

Q1: What is a capacitor and what are its basic components?

Ans: A capacitor is an electrical device that stores electric charge. It consists of two conducting plates separated by an insulating material called a dielectric. The plates, when connected to a voltage source, accumulate equal but opposite charges.

Q2: Explain the laws of capacitors in series and parallel combinations.

Ans: For capacitors in parallel, the effective capacitance is the sum of individual capacitances: $C_{parallel} = C_1 + C_2 + ... + C_n$. For capacitors in series, the reciprocal of the effective capacitance is the sum of the reciprocals of individual capacitances: $\frac{1}{C_{series}} = \frac{1}{C_1} + \frac{1}{C_2} + ... + \frac{1}{C_n}$.

Q3: What is the principle of the repeated charge decay method?

Ans: The repeated charge decay method is based on measuring the rate of discharge of a capacitor through a known high resistance. By measuring the deflections of a ballistic galvanometer at different time intervals during discharge, the capacitance can be calculated using the exponential decay relationship $\frac{\theta_2}{\theta_1} = e^{-\frac{t}{RC}}$.

Q4: Why is a high resistance used in this experiment?

Ans: A high resistance is used to slow down the discharge process, making it easier to measure the time and galvanometer deflections accurately. It ensures that the RC time constant is sufficiently large to allow accurate measurements of the discharge process.

Q5: What is the RC time constant and why is it important in this experiment?

Ans: The RC time constant is the product of resistance (R) and capacitance (C), expressed in seconds. It represents the time taken for the capacitor to discharge to approximately 37% (1/e) of its initial charge. It's important in this experiment because it determines the rate of discharge and helps in selecting appropriate time intervals for measurements.

Q6: What factors affect the capacitance of a capacitor?

Ans: The capacitance of a capacitor depends on the area of the plates (A), the distance between the plates (d), and the dielectric constant (ε) of the material between the plates. The relationship is given by $C = \frac{\varepsilon A}{d}$.

Q7: Why might the experimental values differ from the theoretical values in this experiment?

Ans: Experimental values might differ due to factors such as internal resistance of components, leakage currents in capacitors, measurement errors in time and deflection readings, temperature variations affecting component values, and the non-ideal behavior of real components.

Q8: What is the energy stored in a charged capacitor?

Ans: The energy stored in a charged capacitor is given by $E = \frac{1}{2}CV^2$, where C is the capacitance and V is the voltage across the capacitor.

Q9: How does a ballistic galvanometer work and why is it suitable for this experiment?

Ans: A ballistic galvanometer is designed to measure the total charge that passes through it, rather than the instantaneous current. It works on the principle that the first swing of the galvanometer is proportional to the total charge. This makes it suitable for measuring the discharge of capacitors, where we need to measure the total charge released during discharge.

Q10: What are the applications of capacitors in electronic circuits?

Ans: Capacitors are used in various applications including energy storage, power conditioning, signal coupling and decoupling, filtering, tuning circuits, timing elements in oscillators, sensors, and memory devices in computers.

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