Capacitance Measurement using Ballistic Galvanometer

Determination of Capacitance of a Capacitor by Absolute Method Using Ballistic Galvanometer

1. Aim

To determine the capacitance of a given capacitor using a ballistic galvanometer by the absolute method.

2. Apparatus Used

  • Ballistic galvanometer with lamp and scale arrangement
  • Capacitor of unknown capacitance
  • Battery or DC power supply (typically 2-10V)
  • Resistance box (0-10,000 Ω)
  • Standard capacitor of known capacitance
  • Charge and discharge key (two-way key)
  • One-way key
  • Connecting wires
  • Rheostat
  • Voltmeter (0-10V)

3. Diagram

Circuit diagram for capacitance measurement using ballistic galvanometer

Fig 1: Circuit diagram for the determination of capacitance using a ballistic galvanometer

4. Theory

The ballistic galvanometer is a sensitive instrument designed to measure the quantity of charge. When a capacitor is charged and then discharged through a ballistic galvanometer, the first throw or deflection of the galvanometer is proportional to the quantity of charge passing through it.

When a capacitor of capacitance C is charged to a potential difference V, it stores a charge Q given by:

$Q = CV$

When this charged capacitor is discharged through a ballistic galvanometer, it produces a deflection θ that is proportional to the charge Q:

$\theta = kQ$

Where k is the ballistic constant of the galvanometer.

In the absolute method, we use a standard capacitor of known capacitance (Cs) to determine the ballistic constant k. We then use this constant to find the unknown capacitance (Cx).

For the standard capacitor:

$\theta_s = kQ_s = kC_sV$

For the unknown capacitor:

$\theta_x = kQ_x = kC_xV$

If the same voltage V is used in both cases:

$\frac{\theta_x}{\theta_s} = \frac{C_x}{C_s}$

Therefore:

$C_x = C_s \times \frac{\theta_x}{\theta_s}$

This is the basic principle used in this experiment to determine the unknown capacitance.

5. Formula

The capacitance of the unknown capacitor is given by:

$C_x = C_s \times \frac{\theta_x}{\theta_s}$

Where:

  • $C_x$ = Capacitance of the unknown capacitor (in Farad)
  • $C_s$ = Capacitance of the standard capacitor (in Farad)
  • $\theta_x$ = Deflection of the galvanometer when the unknown capacitor is discharged (in scale divisions)
  • $\theta_s$ = Deflection of the galvanometer when the standard capacitor is discharged (in scale divisions)

Mean deflection for each case is calculated as:

$\theta_{mean} = \frac{\sum \theta}{n}$

Where n is the number of observations.

6. Procedure

  1. Set up the circuit as shown in the diagram. Connect the unknown capacitor (Cx) initially.
  2. Ensure the ballistic galvanometer is properly set up with the lamp and scale arrangement. Adjust the zero position if necessary.
  3. Set the voltage of the DC source to a suitable value (typically 2-5V) using the rheostat. Measure this voltage using the voltmeter.
  4. With key K₂ open, close key K₁ to charge the capacitor.
  5. Open K₁ and then immediately close K₂ to discharge the capacitor through the ballistic galvanometer.
  6. Observe and record the first throw or deflection of the galvanometer (θx).
  7. Repeat steps 4-6 at least five times to get consistent readings for the unknown capacitor.
  8. Replace the unknown capacitor with the standard capacitor (Cs) in the circuit.
  9. Repeat steps 4-7 for the standard capacitor, ensuring that the voltage remains the same as used for the unknown capacitor.
  10. Record the deflections (θs) for the standard capacitor.
  11. Calculate the mean deflections for both capacitors.
  12. Using the formula, calculate the capacitance of the unknown capacitor.

7. Observation Table

Voltage applied (V) = ________ Volts

Capacitance of standard capacitor (Cs) = ________ μF

Observation No. Unknown Capacitor (Cx) Standard Capacitor (Cs)
Deflection (θx)
(scale divisions)		 Direction
(Right/Left)		 Deflection (θs)
(scale divisions)		 Direction
(Right/Left)
1
2
3
4
5
Mean deflection (θx) = Mean deflection (θs) =

8. Calculations

From the observation table:

  • Mean deflection for unknown capacitor (θx) = ________ scale divisions
  • Mean deflection for standard capacitor (θs) = ________ scale divisions
  • Capacitance of standard capacitor (Cs) = ________ μF

Using the formula:

$C_x = C_s \times \frac{\theta_x}{\theta_s}$

Substituting the values:

$C_x = \_\_\_\_\_\_ \times \frac{\_\_\_\_\_\_}{\_\_\_\_\_\_} = \_\_\_\_\_\_ \mu F$

Therefore, the capacitance of the unknown capacitor (Cx) = ________ μF

9. Result

The capacitance of the given capacitor as determined by the absolute method using a ballistic galvanometer is ________ μF.

10. Precautions

  1. The galvanometer should be properly leveled and adjusted to zero before starting the experiment.
  2. The scale should be placed at a suitable distance to get accurate readings.
  3. The same voltage should be used for both the unknown and standard capacitors.
  4. The capacitor should be fully discharged before each observation.
  5. Avoid parallax error while taking readings from the scale.
  6. The keys should be operated in the correct sequence to ensure proper charging and discharging.
  7. The connection wires should have low resistance and good insulation.
  8. Ensure there is no external magnetic field that might affect the galvanometer readings.
  9. Avoid vibrations near the galvanometer setup.
  10. The standard capacitor should have a reliable and accurately known value.

11. Viva Voice Questions

  1. Q: What is a ballistic galvanometer?
    A: A ballistic galvanometer is a sensitive moving coil galvanometer designed to measure the quantity of charge or the magnetic flux. Unlike a regular galvanometer that measures current, a ballistic galvanometer gives a deflection proportional to the total charge passed through it in a short time interval.
  2. Q: Why is it called the "absolute method"?
    A: It is called the absolute method because we use a standard capacitor of known capacitance as a reference to determine the unknown capacitance. The method directly compares the deflections produced by the two capacitors under identical conditions.
  3. Q: What is the principle of the ballistic galvanometer?
    A: The principle states that the first throw or deflection of the ballistic galvanometer is directly proportional to the quantity of charge passing through it, provided the discharge occurs in a time much shorter than the time period of oscillation of the galvanometer coil.
  4. Q: Why are repeated observations taken in this experiment?
    A: Repeated observations are taken to minimize random errors and to get a more reliable mean value for the deflections. This increases the accuracy of the final result.
  5. Q: What would happen if the capacitor is not fully discharged between observations?
    A: If the capacitor is not fully discharged between observations, residual charge will remain, leading to incorrect measurements in subsequent trials. This would result in inconsistent deflection readings and an inaccurate calculation of capacitance.
  6. Q: How does the voltage affect the deflection in the galvanometer?
    A: The deflection in the galvanometer is directly proportional to the charge, which is the product of capacitance and voltage (Q = CV). Therefore, increasing the voltage will increase the deflection proportionally, assuming the capacitance remains constant.
  7. Q: What are the sources of error in this experiment?
    A: Sources of error include parallax error in reading deflections, inaccuracy in the value of the standard capacitor, leakage current in capacitors, thermal drift in the galvanometer, external magnetic fields affecting the galvanometer, and inconsistent voltage supply.
  8. Q: Why is it important to use the same voltage for both capacitors?
    A: Using the same voltage ensures that the only variable affecting the deflection is the capacitance. This allows for a direct comparison between the unknown and standard capacitors using the ratio of their deflections.
  9. Q: What is the significance of the ballistic constant?
    A: The ballistic constant relates the charge passed through the galvanometer to the resulting deflection (θ = kQ). It is a characteristic of the specific galvanometer being used and is determined by the galvanometer's physical properties.
  10. Q: How would you modify this experiment to measure very small capacitances?
    A: For very small capacitances, you could use a higher voltage to increase the deflection, use a more sensitive galvanometer, employ a standard capacitor of appropriate small value for comparison, or use a series combination of multiple identical unknown capacitors and then calculate the individual capacitance.

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