Determination of Galvanometer Resistance by Half-Deflection Method
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1. Aim
To determine the resistance of a given galvanometer by the half-deflection method and to calculate its figure of merit.
2. Apparatus Used
- A moving coil galvanometer
- A low resistance shunt box
- A resistance box (0-10,000 Ω)
- A battery (1.5V or 2V)
- A rheostat
- A one-way key
- Connecting wires
- A meter scale
3. Diagram
Fig.1: Circuit diagram for determining galvanometer resistance by half-deflection method
4. Theory
The half-deflection method is based on the principle that when a galvanometer is connected in parallel with a suitable resistance (shunt), the current through the galvanometer reduces, resulting in a decreased deflection.
When a current flows through a galvanometer, the deflection produced is directly proportional to the current passing through it:
$\theta \propto I$
When a galvanometer of resistance G is connected in a circuit with a cell of emf E and a resistance R, the current through the galvanometer is:
$I = \frac{E}{R+G}$
When a shunt S is connected across the galvanometer, the current through the galvanometer becomes:
$I' = \frac{E}{R+\frac{G \times S}{G+S}}$
According to the half-deflection method, if S is adjusted such that the deflection becomes half of the original deflection, then:
$I' = \frac{I}{2}$
This means:
$\frac{E}{R+\frac{G \times S}{G+S}} = \frac{E}{2(R+G)}$
Solving this equation:
\begin{align}
R+\frac{G \times S}{G+S} &= 2(R+G) \\
\frac{G \times S}{G+S} &= 2R+2G-R \\
\frac{G \times S}{G+S} &= R+2G
\end{align}
Further simplification leads to:
$G = S$
Therefore, when the deflection is reduced to half by connecting a shunt across the galvanometer, the value of the shunt resistance equals the resistance of the galvanometer.
5. Formula
1. Resistance of the galvanometer (G):
$G = S$ (shunt resistance when deflection becomes half)
2. Figure of merit (k):
$k = \frac{I}{n} = \frac{E}{(R+G) \times n}$
where:
- I is the current through the galvanometer = $\frac{E}{R+G}$
- n is the deflection in divisions
- E is the emf of the battery
- R is the resistance in the circuit
6. Procedure
- Connect the circuit as shown in the diagram without connecting the shunt resistance.
- Close the key K and adjust the resistance R such that a suitable deflection (preferably near full scale) is obtained on the galvanometer. Record this deflection (n₁).
- Open the key and note the zero error, if any.
- Connect a suitable shunt resistance S across the galvanometer.
- Close the key again and adjust the value of S until the deflection becomes exactly half of the original value (n₂ = n₁/2).
- Record the value of the shunt resistance S when half-deflection is achieved.
- Calculate the resistance of the galvanometer using G = S.
- For determining the figure of merit, calculate the current through the galvanometer: $I = \frac{E}{R+G}$
- Calculate the figure of merit using k = I/n₁.
- Repeat the experiment for different values of R and calculate the average value of G and k.
7. Observation Table
Table 1: Determination of Galvanometer Resistance
| S.No. | Circuit resistance (R) Ω | Original deflection (n₁) divisions | Shunt resistance for half-deflection (S) Ω | Galvanometer resistance (G = S) Ω |
|---|---|---|---|---|
| 1. | ||||
| 2. | ||||
| 3. | ||||
| 4. | ||||
| 5. |
Mean value of galvanometer resistance (G) = ________ Ω
Table 2: Determination of Figure of Merit
| S.No. | Circuit resistance (R) Ω | Galvanometer resistance (G) Ω | Current (I = E/(R+G)) A | Deflection (n₁) divisions | Figure of merit (k = I/n₁) A/division |
|---|---|---|---|---|---|
| 1. | |||||
| 2. | |||||
| 3. | |||||
| 4. | |||||
| 5. |
Mean value of figure of merit (k) = ________ A/division
8. Calculations
1. The resistance of the galvanometer is given by:
$G = S$
2. The current through the galvanometer is given by:
$I = \frac{E}{R+G}$
where E is the emf of the battery used (in volts)
3. The figure of merit is calculated as:
$k = \frac{I}{n_1}$ (A/division)
Sample calculation for the first reading:
- Let R = 1000 Ω
- Original deflection (n₁) = 20 divisions
- Shunt resistance for half-deflection (S) = 50 Ω
- EMF of battery (E) = 1.5 V
Therefore:
- Galvanometer resistance (G) = S = 50 Ω
- Current through galvanometer (I) = $\frac{1.5}{1000+50} = \frac{1.5}{1050} = 0.00143$ A
- Figure of merit (k) = $\frac{0.00143}{20} = 7.14 \times 10^{-5}$ A/division
9. Result
- The resistance of the given galvanometer is ________ Ω.
- The figure of merit of the given galvanometer is ________ A/division.
10. Precautions
- All connections should be tight and clean.
- The battery should have sufficient emf and should be in good condition.
- The galvanometer should be placed on a level surface, free from vibrations.
- The galvanometer should be properly adjusted to zero before taking readings.
- The resistance box should be handled carefully, and plugs should be inserted firmly.
- The key should be pressed gently to avoid any jerk.
- The current passing through the galvanometer should not exceed its maximum safe limit.
- Parallax error should be avoided while taking readings from the galvanometer scale.
- The resistance R should be adjusted to get a suitable deflection (about 3/4 of the full scale).
11. Sources of Error
- Zero error in the galvanometer.
- Error due to improper adjustment of the resistance box.
- Error due to improper connections or loose contacts.
- Error due to internal resistance of the battery.
- Error due to variations in battery voltage during the experiment.
- Error due to temperature variations affecting the resistance values.
- Parallax error while reading the galvanometer scale.
- Error due to fluctuations in the magnetic field around the galvanometer.
- Error due to the heating effect of current in the circuit components.
12. Viva Voce Questions
Q: What is a galvanometer?
A: A galvanometer is a sensitive electromagnetic device used to detect and measure small electric currents.
Q: What is the principle of a moving coil galvanometer?
A: A moving coil galvanometer works on the principle that when a current-carrying coil is placed in a magnetic field, it experiences a torque which causes the coil to rotate. The deflection of the coil is directly proportional to the current passing through it.
Q: What is the half-deflection method?
A: The half-deflection method is a technique used to determine the resistance of a galvanometer by connecting a shunt resistance across it so that the deflection becomes half of its original value. At this point, the shunt resistance equals the galvanometer resistance.
Q: What is a shunt in the context of galvanometers?
A: A shunt is a low-resistance conductor connected in parallel with a galvanometer to allow a portion of the current to bypass the galvanometer, thereby protecting it from excessive current and extending its range of measurement.
Q: What is the figure of merit of a galvanometer?
A: The figure of merit of a galvanometer is the current required to produce a deflection of one division on its scale. It is expressed in amperes per division.
Q: How can you convert a galvanometer into an ammeter?
A: A galvanometer can be converted into an ammeter by connecting a low-resistance shunt in parallel with it. The shunt diverts most of the current around the galvanometer, allowing it to measure larger currents.
Q: How can you convert a galvanometer into a voltmeter?
A: A galvanometer can be converted into a voltmeter by connecting a high-resistance in series with it. This limits the current flowing through the galvanometer to a safe value proportional to the voltage being measured.
Q: Why is the half-deflection method preferred over other methods for determining galvanometer resistance?
A: The half-deflection method is preferred because it is simple, accurate, and does not require knowledge of the emf of the battery or the exact value of the external resistance in the circuit.
Q: What factors affect the sensitivity of a galvanometer?
A: The sensitivity of a galvanometer is affected by factors such as the strength of the magnetic field, the number of turns in the coil, the area of the coil, the torsional constant of the suspension, and the resistance of the galvanometer.
Q: What is the significance of the figure of merit in practical applications?
A: The figure of merit is significant because it helps in calibrating the galvanometer and converting its deflection into the corresponding current value. It is essential for using the galvanometer as a measuring instrument.