Understanding Ohm's law and ohmic conductors

Ohmic conductors, at a steady temperature, obey Ohm's law.

The potential difference across a component is directly proportional to the current flowing through it and the resistance of the component.

$V = IR$

Where:

• V = Voltage (V)
• I = Current (A)
• R = resistance (Ω)

Key points:

• Doubling the voltage results in a doubled current.
• The resistance remains constant under these conditions.
• This law does not apply to non-ohmic materials at varying temperatures.

Three key factors influencing resistance

The resistance (R) of a conductor is influenced by:

• Length (L) - The longer the wire, the greater its resistance.
• Area (A) - Wires with a larger cross-sectional area have reduced resistance.
• Resistivity ($\rho$) - This is a property unique to the material, which varies with temperature.

Resistivity is defined as the resistance encountered in a material that is 1 metre long and has a cross-sectional area of 1 square metre. Its units are ohm-metres (Ωm).

The formula linking these factors is:

$R = \frac{\rho L}{A}$

Where:

R = resistance ($\Omega$)

$\rho$ = resistivity ($\Omega$ m$)$

L = length (m)

A = area (m$^2$)

Worked example - Calculating the resistance of a copper wire

We will calculate the resistance of a copper wire that is 2 metres long and 1 mm in diameter.

The resistivity of copper is 1.72 x 10^-8 $\Omega$ m.

Step 1: Calculate the Cross-sectional Area (A)

$A = \pi \times r^2$

$A = \pi \times 0.0005^2 = 7.85 \times 10^{-7} \text{ m}^2$

Step 2: Formula

$\text{R = }\frac{\rho\text{ l}}{\text{A}}$

Step 3: Substitution and correct evaluation

$R = \frac{1.72 \times 10^{-8}\times 2 }{7.85 \times 10^{-7} }=0.044\text{ } \Omega$

Determining the resistivity of a wire experimentally

To find out the resistivity of a wire:

1. Measure the diameter of the wire using a micrometer and calculate its cross-sectional area: $A = \frac{\pi d^2}{4}$ .
2. Set up a circuit with the wire, power supply, ammeter and voltmeter
3. Record the voltage and current readings.
4. Vary the length of the wire and record the new voltage and current readings.
5. Calculate the resistance for each length using: $R = \frac{V}{I}$
6. Create a graph plotting length against resistance.
7. The gradient of this graph equals $\frac{\rho}{A}$
8. Multiply the gradient by A to obtain the wire's resistivity.