Verification of Laws of Combination of Resistances
1. AIM
To verify the laws of combination of resistances (series and parallel) using a metre bridge and to determine the resultant resistance in each case.
2. APPARATUS USED
- Metre bridge (slide wire bridge)
- Galvanometer
- Resistance box (R.B.)
- Jockey
- Two unknown resistances (R₁ and R₂)
- One-way key
- Connecting wires
- Sandpaper
- Battery or power supply (2V)
- Commutator (optional)
3. DIAGRAM
Fig. 1: Circuit diagram for verification of laws of combination of resistances using a metre bridge
4. THEORY
The metre bridge works on the principle of Wheatstone bridge. The metre bridge consists of a one-metre long uniform wire of manganin or constantan fixed on a wooden board with copper strips at its ends. The wire is stretched along a scale graduated in centimetres or millimetres.
When the Wheatstone bridge is balanced (no current flows through the galvanometer), the ratio of resistances in the four arms follows the relation:
\[ \frac{P}{Q} = \frac{R}{S} \]
In a metre bridge, the wire AB acts as two arms of the Wheatstone bridge. If l is the length of the wire from end A to the jockey position (null point), and (100-l) is the remaining length, then:
\[ \frac{P}{Q} = \frac{l}{(100-l)} \]
where l is in centimetres.
For a balanced bridge, if R is the known resistance connected to gap 1, and X is the unknown resistance connected to gap 2, then:
\[ \frac{R}{X} = \frac{l}{(100-l)} \]
\[ X = R\times\frac{(100-l)}{l} \]
Laws of Combination of Resistances:
- Series Combination: When resistors are connected in series, the equivalent resistance (Rs) is the sum of individual resistances:
\[ R_s = R_1 + R_2 + R_3 + ... + R_n \]
- Parallel Combination: When resistors are connected in parallel, the reciprocal of the equivalent resistance (Rp) is the sum of reciprocals of individual resistances:
\[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} + \frac{1}{R_3} + ... + \frac{1}{R_n} \]
5. FORMULA
- For determining the resistance (X):
\[ X = R \times \frac{(100-l)}{l} \]
- For series combination:
\[ R_s = R_1 + R_2 \]
- For parallel combination:
\[ \frac{1}{R_p} = \frac{1}{R_1} + \frac{1}{R_2} \] \[ R_p = \frac{R_1 \times R_2}{R_1 + R_2} \]
6. PROCEDURE
- Set up the apparatus:
- Place the metre bridge on a horizontal surface.
- Clean all connections using sandpaper to ensure good electrical contact.
- Connect the circuit as shown in the diagram.
- Determination of Resistance R₁:
- Connect the resistance R₁ in gap 2 of the metre bridge.
- Place a suitable resistance R from the resistance box in gap 1.
- Close the key and move the jockey along the wire until the galvanometer shows zero deflection (null point).
- Record the position of the jockey (l₁) and the value of R used.
- Determine the value of R₁ using the formula: R₁ = R × (100-l₁)/l₁
- Determination of Resistance R₂:
- Replace R₁ with R₂ in gap 2.
- Adjust R if necessary to obtain a balance point within the middle portion of the bridge wire.
- Find the null point position l₂ and record it.
- Calculate R₂ using the formula: R₂ = R × (100-l₂)/l₂
- Series Combination:
- Connect R₁ and R₂ in series and place this combination in gap 2.
- Find the new balance point l₃ and record it.
- Calculate the series combination resistance Rs = R × (100-l₃)/l₃
- Compare this with the theoretical value Rs = R₁ + R₂
- Parallel Combination:
- Connect R₁ and R₂ in parallel and place this combination in gap 2.
- Find the new balance point l₄ and record it.
- Calculate the parallel combination resistance Rp = R × (100-l₄)/l₄
- Compare this with the theoretical value Rp = (R₁ × R₂)/(R₁ + R₂)
7. OBSERVATION TABLE
Table 1: Determination of Individual Resistances
| S.No. | Resistance in gap 1 (R) (Ω) | Position of null point (l) (cm) | Unknown resistance (X = R×(100-l)/l) (Ω) |
|---|---|---|---|
| 1. | R | l₁ | R₁ = R×(100-l₁)/l₁ |
| 2. | R | l₂ | R₂ = R×(100-l₂)/l₂ |
Table 2: Verification of Series Combination Law
| S.No. | Resistance in gap 1 (R) (Ω) | Position of null point for series combination (l₃) (cm) | Observed series resistance Rs = R×(100-l₃)/l₃ (Ω) | Calculated series resistance R₁+R₂ (Ω) | Percentage error (%) |
|---|---|---|---|---|---|
| 1. | R | l₃ | Rs | R₁+R₂ | [(R₁+R₂)-Rs]/(R₁+R₂)×100 |
Table 3: Verification of Parallel Combination Law
| S.No. | Resistance in gap 1 (R) (Ω) | Position of null point for parallel combination (l₄) (cm) | Observed parallel resistance Rp = R×(100-l₄)/l₄ (Ω) | Calculated parallel resistance (R₁×R₂)/(R₁+R₂) (Ω) | Percentage error (%) |
|---|---|---|---|---|---|
| 1. | R | l₄ | Rp | (R₁×R₂)/(R₁+R₂) | [(R₁×R₂)/(R₁+R₂)-Rp]/[(R₁×R₂)/(R₁+R₂)]×100 |
8. CALCULATIONS
- Individual Resistances:
\[ R_1 = R \times \frac{(100-l_1)}{l_1} \] \[ R_2 = R \times \frac{(100-l_2)}{l_2} \]
- Series Combination:
\[ \text{Observed value: } R_s = R \times \frac{(100-l_3)}{l_3} \] \[ \text{Theoretical value: } R_s \text{ (theoretical)} = R_1 + R_2 \] \[ \text{Percentage error} = \frac{(R_1+R_2)-R_s}{(R_1+R_2)} \times 100 \]
- Parallel Combination:
\[ \text{Observed value: } R_p = R \times \frac{(100-l_4)}{l_4} \] \[ \text{Theoretical value: } R_p \text{ (theoretical)} = \frac{R_1 \times R_2}{R_1 + R_2} \] \[ \text{Percentage error} = \frac{[\frac{R_1 \times R_2}{R_1 + R_2}]-R_p}{[\frac{R_1 \times R_2}{R_1 + R_2}]} \times 100 \]
9. RESULT
- The measured values of the individual resistances are:
- R₁ = _____ Ω
- R₂ = _____ Ω
- For Series Combination:
- Observed value: Rs = _____ Ω
- Theoretical value: R₁ + R₂ = _____ Ω
- Percentage error: _____ %
- The law of series combination of resistances is verified with an error of _____ %.
- For Parallel Combination:
- Observed value: Rp = _____ Ω
- Theoretical value: (R₁ × R₂)/(R₁ + R₂) = _____ Ω
- Percentage error: _____ %
- The law of parallel combination of resistances is verified with an error of _____ %.
10. PRECAUTIONS
- The connections should be tight and clean.
- The jockey should make gentle contact with the wire to avoid damaging it.
- The resistance box plugs should be tight and clean.
- The null point should be determined carefully by observing the zero deflection in the galvanometer.
- The metre bridge wire should be uniform and free from kinks.
- The resistance in gap 1 should be chosen so that the null point falls within the middle portion (30-70 cm) of the wire for better accuracy.
- The battery emf should not be too high to avoid heating of the wire.
- The jockey should not be kept in contact with the wire for a long time to avoid heating effects.
- The galvanometer should be protected from external magnetic fields.
- The experiment should be performed in a vibration-free environment.
11. SOURCES OF ERROR
- The metre bridge wire may not be perfectly uniform in cross-section throughout its length.
- The end corrections due to resistance of thick copper strips are neglected.
- The resistance of connecting wires may affect the measurements.
- Thermal emf may be generated at junctions of dissimilar metals.
- Heating effect due to current flow may change the resistance values during the experiment.
- Improper contact of the jockey with the wire may cause reading errors.
- Parallax error while reading the position of the jockey.
- Zero error in the metre scale.
- Improper balancing of the Wheatstone bridge.
- External electromagnetic interference affecting the galvanometer readings.
12. VIVA VOICE QUESTIONS
Q1: What is the principle of a metre bridge?
A: The metre bridge works on the principle of a Wheatstone bridge, where at balance point, the ratio of resistances in the four arms follows the relation P/Q = R/S.
Q2: Why is the metre bridge wire made of manganin or constantan?
A: Manganin or constantan have high resistivity, low temperature coefficient of resistance, and do not oxidize easily, making them suitable for precise measurements over time.
Q3: Why should the null point be preferably in the middle portion of the metre bridge?
A: The sensitivity of the bridge is maximum when the null point is at the center. As we move away from the center, the sensitivity decreases, leading to less accurate measurements.
Q4: What is the effect of temperature on the resistance measurement?
A: Increase in temperature increases the resistance of most conductors (except semiconductors), which can affect the accuracy of measurements if significant heating occurs.
Q5: Why is the jockey not kept in contact with the wire for a long time?
A: To avoid heating of the wire due to prolonged current flow, which would change its resistance and affect the measurements.
Q6: What would happen if the wire of the metre bridge is not uniform?
A: Non-uniformity in wire would cause errors in measurement as the resistance per unit length would not be constant throughout the wire.
Q7: Why should the contact points be clean and tight?
A: To reduce contact resistance, which would otherwise introduce errors in the measurements.
Q8: What is end correction in a metre bridge?
A: End correction accounts for the additional resistance at the ends of the bridge due to the thick copper strips and contact resistances.
Q9: Why is a high resistance galvanometer preferred in this experiment?
A: A high resistance galvanometer draws less current, minimizing the loading effect on the bridge circuit and improving accuracy.
Q10: Can we use this method to measure very high or very low resistances? Why or why not?
A: It's difficult to measure very high or very low resistances accurately with a metre bridge. For very high resistances, the balance point would be too close to one end, and for very low resistances, it would be difficult to distinguish small changes in position.