1. Aim

To determine the resistance per unit length (per cm) of a given wire by plotting a graph of potential difference versus current.

2. Apparatus Used

• A wire of uniform cross-section (typically nichrome or constantan wire)
• Meter scale
• Battery or power supply (0-12V DC)
• Ammeter (0-2A)
• Voltmeter (0-5V)
• Rheostat (variable resistor)
• Connecting wires
• One-way key (switch)
• Two crocodile clips or jockey (movable contacts)
• A wooden board with graph paper
• Drawing pins

3. Diagram

4. Theory

When a current flows through a conductor, the potential difference (V) across it is directly proportional to the current (I) flowing through it, according to Ohm's law:

Where:

• $V$ is the potential difference across the conductor (in volts)
• $I$ is the current flowing through the conductor (in amperes)
• $R$ is the resistance of the conductor (in ohms)

The resistance ($R$) of a uniform wire depends on its length ($L$), cross-sectional area ($A$), and the material's resistivity ($\rho$):

For a wire of uniform cross-section, the resistance is directly proportional to its length:

Where:

• $r$ is the resistance per unit length ($\Omega/cm$)
• $L$ is the length of the wire (cm)

By measuring the potential difference ($V$) across a specific length of wire for various currents ($I$) and plotting a V-I graph, we can determine the resistance of that length of wire from the slope of the graph. By dividing this resistance by the length, we obtain the resistance per unit length.

5. Formula

The resistance of the wire is calculated using Ohm's law:

The resistance per unit length ($r$) is given by:

Where:

• $r$ is the resistance per unit length ($\Omega/cm$)
• $R$ is the resistance of the wire ($\Omega$)
• $V$ is the potential difference across the wire (volts)
• $I$ is the current flowing through the wire (amperes)
• $L$ is the length of the wire (cm)

From the V-I graph, the slope gives the resistance:

Therefore, resistance per unit length:

6. Procedure

1. Arrange the apparatus as shown in the circuit diagram.
2. Stretch the experimental wire along the meter scale and fix it firmly on the board.
3. Note the total length ($L$) of the wire in centimeters.
4. Keep the key (K) open and set the rheostat to maximum resistance position.
5. Close the key (K) and adjust the rheostat to get a suitable reading on the ammeter (e.g., 0.1A).
6. Note the reading of current ($I$) on the ammeter and the corresponding potential difference ($V$) on the voltmeter.
7. Change the position of the rheostat to get different values of current and note the corresponding voltmeter readings.
8. Take at least 6-8 different readings by varying the current.
9. Open the key after taking all readings.
10. Record all observations in the observation table.
11. Plot a graph with potential difference ($V$) on the Y-axis and current ($I$) on the X-axis.
12. Determine the slope of the graph, which represents the resistance ($R$) of the wire.
13. Calculate the resistance per unit length ($r$) by dividing the resistance by the length of the wire.

7. Observation Table

Wire material: ____________

Total length of wire (L): ____________ cm

Diameter of wire: ____________ mm

Cross-sectional area (A): ____________ mm²

S.No.	Current (I) Amperes	Potential Difference (V) Volts
1
2
3
4
5
6
7
8

8. Calculations

1. Plot a graph with Potential Difference ($V$) on the Y-axis and Current ($I$) on the X-axis.
2. Draw the best-fit straight line through the plotted points.
3. Calculate the slope of the graph:

   $\text{Slope} = \frac{\Delta V}{\Delta I} = \frac{V_2 - V_1}{I_2 - I_1}$

   where $(I_1, V_1)$ and $(I_2, V_2)$ are two points on the straight line.

4. The slope gives the resistance ($R$) of the wire in ohms.
5. Calculate the resistance per unit length:

   $r = \frac{R}{L} = \frac{\text{Slope}}{L} \, (\Omega/\text{cm})$

If the slope of the V-I graph is 2.5 $\Omega$ and the length of the wire is 100 cm:

9. Result

The resistance per unit length (per cm) of the given wire is ____________ $\Omega/\text{cm}$.

10. Precautions

1. The wire should be of uniform cross-section and free from kinks or twists.
2. All connections should be tight and clean to minimize contact resistance.
3. The key should be closed only while taking readings to prevent heating of the wire.
4. The rheostat should be adjusted slowly to avoid sudden changes in current.
5. The wire should be stretched straight without any sag.
6. The ammeter and voltmeter should be of appropriate range and properly calibrated.
7. Zero error of the measuring instruments should be checked and accounted for.
8. Readings should be taken quickly to minimize heating effects.
9. Avoid parallax error while reading the instruments.
10. The experiment should be performed at constant temperature.

11. Sources of Error

1. The wire may not have perfectly uniform cross-section throughout its length.
2. Contact resistance at the junctions may affect the readings.
3. Heating of the wire due to current flow may change its resistance during the experiment.
4. Parallax error while reading the measuring instruments.
5. Zero error in the ammeter and voltmeter.
6. Non-uniformity in the temperature of the wire.
7. The wire's length measurement may have slight errors.
8. Fluctuations in the battery or power supply voltage.
9. Internal resistance of the ammeter and battery may affect the readings.
10. Loose connections in the circuit can cause inconsistent readings.

12. Viva Voice Questions

Q: What is Ohm's law?

A: Ohm's law states that the current flowing through a conductor is directly proportional to the potential difference across it, provided the physical conditions (like temperature) remain constant. Mathematically, $V = IR$.

Q: Why do we plot a graph between V and I instead of just using a single reading?

A: We plot a graph to verify that the wire follows Ohm's law (linear relationship between V and I) and to obtain a more accurate value of resistance by considering multiple readings, which minimizes random errors.

Q: How does the resistance of a wire vary with its length?

A: The resistance of a wire is directly proportional to its length. If the length doubles, the resistance also doubles.

Q: How does the resistance of a wire vary with its cross-sectional area?

A: The resistance of a wire is inversely proportional to its cross-sectional area. If the area doubles, the resistance halves.

Q: What is resistivity? How does it differ from resistance?

A: Resistivity is a material property that indicates how strongly the material opposes the flow of electric current. Resistance depends on both the material (resistivity) and the physical dimensions (length and cross-sectional area) of the conductor.

Q: Why does the resistance of a wire increase with temperature?

A: In most metals, as temperature increases, the thermal vibrations of atoms increase, which impedes the flow of electrons, thus increasing resistance.

Q: What would happen to the V-I graph if the wire gets heated significantly during the experiment?

A: The graph would deviate from a straight line, showing a curve with increasing slope (increasing resistance) at higher currents due to the heating effect.

Q: Why is a rheostat used in this experiment?

A: A rheostat is used to vary the current in the circuit in a controlled manner so that we can take readings at different current values.

Q: What would happen if the wire is not of uniform cross-section?

A: The resistance per unit length would not be constant throughout the wire, leading to errors in the measurement and potentially a non-linear V-I relationship.

Q: How would you calculate the resistivity of the wire from this experiment?

A: Resistivity ($\rho$) can be calculated using the formula $\rho = R \times A/L$, where $R$ is the resistance, $A$ is the cross-sectional area, and $L$ is the length of the wire.