Resistivity of a wire by plotting a V-I graph (Class 12 practical)

Aim

To determine resistivity of two / three wires by plotting a graph for potential difference versus current.

Apparatus and material required

• •Two or three resistance wires (each about 10 Ω or less)
• •Battery eliminator (0 to 3 V) or two dry cells of 1.5 V
• •Voltmeter (0 to 5 V)
• •Milliammeter (0 to 500 mA)
• •Rheostat
• •Plug key
• •Connecting wires and a piece of sandpaper
• •Metre scale
• •Screw gauge (to find the diameter of each wire)

Principle

Resistivity of a wire from the V-I graph: formulas and key facts

Ohm's law

$V=RI$

Holds while the temperature of the wire stays the same.

Resistance from the graph

$R=\text{slope of }V\text{-}I\text{ graph}=\frac{\Delta V}{\Delta I}$

If you plot I on the y-axis instead, the slope is 1/R.

Resistivity

$\rho =\frac{RA}{l}=\frac{R\pi r^2}{l}=\frac{R\pi d^2}{4l}$

SI unit: Ω m. Take r in metres, not millimetres.

Resistance per unit length

$\frac{R}{l}$

Unit Ω m⁻¹.

Fractional error

$\frac{\Delta R}{R}=\frac{\Delta V}{V}+\frac{\Delta I}{I}$

ΔV and ΔI are the least counts of the voltmeter and milliammeter.

Key facts

• •Voltmeter in parallel with the wire
• •Ammeter (milliammeter) in series
• •Current enters each meter at its + terminal
• •Insert the key only while reading
• •Graph: straight line through the origin
• •Resistivity depends on material and temperature, not on length or area

Procedure

1. 1Clean the ends of the connecting wires with sandpaper to remove the insulating coating.
2. 2Measure the length $l$ of the first wire between the two voltmeter terminals with a metre scale, and its diameter with a screw gauge at three or four places, in two perpendicular directions at each place.
3. 3Connect the wire, rheostat, battery, key and milliammeter in series, and the voltmeter in parallel across the wire. Check that current enters each meter at its positive terminal.
4. 4Check that the pointers of the milliammeter and voltmeter are on zero. If not, set them with the screw at the base of the pointer. Note the range and least count of each meter.
5. 5Insert the key and slide the rheostat contact to one end so that the current is the least. Note the milliammeter and voltmeter readings.
6. 6Remove the key and let the wire cool. Insert it again, move the rheostat contact a little to increase the voltage, and note both readings.
7. 7Repeat for four or five more settings of the rheostat, so you have five or six pairs of readings.
8. 8Repeat the whole experiment with the second (and third) wire.
9. 9For each wire, plot V on the y-axis against I on the x-axis, draw the best straight line through the origin, and find its slope.

Observation table

Draw the table like this in your record. The numbers are sample readings to show how it is filled in; use your own.

Calculation

1. 1.The best-fit line passes very close to the origin (least-squares intercept about 0.02 V, smaller than the least count). Two points read off the line: (50 mA, 0.42 V) and (300 mA, 2.42 V).
2. 2.$R=\frac{\Delta V}{\Delta I}=\frac{2.42-0.42}{(300-50)\times 10^{-3}}=\frac{2.00}{0.250}=8.0\Omega$
3. 3.Resistance per unit length $=\frac{R}{l}=\frac{8.0}{0.500}=16.0\Omega {\text{m}}^{-1}$
4. 4.Area of cross-section $A=\pi r^2=3.1416\times (1.0\times 10^{-4}{)}^2=3.14\times 10^{-8}{\text{m}}^2$
5. 5.$\rho =\frac{RA}{l}=\frac{8.0\times 3.14\times 10^{-8}}{0.500}=5.03\times 10^{-7}\approx 5.0\times 10^{-7}\Omega \text{m}$
6. 6.This lies in the range NCERT's data section gives for constantan (44 to 52 × 10⁻⁸ Ω m). Repeat steps 1 to 5 for wire 2 (and wire 3).

Result

1. The V-I graph of each wire is a straight line through the origin, so V ∝ I (Ohm's law holds). 2. Resistivity of wire 1 = ____ Ω m; of wire 2 = ____ Ω m (sample: wire 1, R = 8.0 Ω, ρ = 5.0 × 10⁻⁷ Ω m).

Precautions

• →Connect the voltmeter in parallel with the wire and the milliammeter in series, with current entering each at the positive terminal.
• →Insert the key only while taking readings, so the wire does not heat up and change its resistance.
• →Remove zero error in the meters with the adjusting screw, and in the screw gauge by applying the zero correction.
• →Measure the length of the wire only between the voltmeter terminals.
• →Measure the diameter at several places along the wire, in two perpendicular directions at each place.

Sources of error

• •The wire may not have a uniform area of cross-section.
• •Lengths of wire wound round the terminals, if included, make the measured length wrong.
• •The wire warms up if current flows for long, which increases its resistance.
• •The diameter is small, so a small error in it gives a large error in the area (r appears squared).

Viva questions with answers

Say your answer out loud first, then open the card. Examiners want one or two clear sentences, not a paragraph.

1. 1.State Ohm's law.Show answer

   The current through a conductor is directly proportional to the potential difference across its ends, provided its physical conditions such as temperature stay the same.

2. 2.What is the shape of the V-I graph for a metallic wire, and what does its slope give?Show answer

   A straight line through the origin. With V on the y-axis and I on the x-axis, the slope is the resistance R. If I is on the y-axis, the slope is 1/R.

3. 3.Define resistivity. What is its SI unit?Show answer

   Resistivity is the resistance of a wire of the material of unit length and unit area of cross-section; $\rho =RA/l$. Its SI unit is the ohm metre (Ω m).

4. 4.On what factors does the resistivity of a wire depend?Show answer

   Only on the material and its temperature. It does not depend on the length or the area of cross-section of the wire.

5. 5.If the wire is stretched to double its length, what happens to its resistance and resistivity?Show answer

   The volume stays the same, so the area halves. $R=\rho l/A$ becomes four times. The resistivity does not change, because the material is the same.

6. 6.Why is the voltmeter connected in parallel and the ammeter in series?Show answer

   A voltmeter measures the potential difference between two points, so it goes across them; it has a high resistance and draws little current. An ammeter must carry the whole current, so it goes in series; it has a low resistance.

7. 7.Why do we insert the key only while taking a reading?Show answer

   If current flows for a long time, the wire heats up and its resistance increases, so the readings would no longer follow V ∝ I.

8. 8.What is the use of the rheostat in this circuit?Show answer

   It changes the current in the circuit, so we can take several pairs of V and I readings.

9. 9.Why are copper wires used for the connections?Show answer

   Copper has a very low resistivity, so the connecting wires have negligible resistance and almost no potential drop across them.

10. 10.Why do we measure the diameter at several places and in two perpendicular directions?Show answer

    The wire may not be perfectly uniform or perfectly round. Averaging reduces this error, and it matters because the diameter enters the area as a square.

11. 11.Name a device that does not obey Ohm's law.Show answer

    A p-n junction diode (also a thyristor). Its V-I graph is not a straight line; such devices are called non-ohmic.

12. 12.Which material would you choose for a standard resistance coil, and why?Show answer

    Constantan or manganin: they have a high resistivity and a very small temperature coefficient, so their resistance hardly changes with temperature.

Viva practice: these questions one at a time, with the answer hidden until you ask, and a tally of what to revise.

Common mistakes in the practical exam

The slips examiners see most often, and what to write instead.

• ✗Mistake: Finding R by averaging the V/I values and calling it the graph result.

  The aim says to plot a graph; the examiner looks for the slope.

  ✓Do this: Draw the best straight line, mark two points far apart on it, and show ΔV/ΔI on the graph sheet.

• ✗Mistake: Using the diameter in millimetres in ρ = Rπr²/l.

  The answer comes out a million times too big.

  ✓Do this: Convert r to metres first: 0.10 mm = 1.0 × 10⁻⁴ m.

• ✗Mistake: Using the diameter as the radius.

  The area becomes four times too large.

  ✓Do this: Halve the mean diameter after the zero correction, then square it.

• ✗Mistake: Forgetting the zero correction of the screw gauge.

  ✓Do this: Note the zero error before measuring, and write the corrected mean diameter in the table.

• ✗Mistake: Not starting the graph at the origin or choosing a scale that uses a quarter of the sheet.

  ✓Do this: Start both axes at zero, and choose a scale that spreads the readings over most of the sheet.

Resistivity of a wire from the V-I graph: frequently asked questions