Resistance

Section: Electricity  |  Syllabus: Cambridge AS Level Physics (9702)

Resistance

Resistance is defined as the ratio of the potential difference across a component to the current flowing through it.

Resistance R = (V/I)
where R = resistance (Ω), V = potential difference (V), I = current (A)

Resistance is measured in ohms (Ω). One ohm is the resistance when a p.d. of one volt causes a current of one ampere.

Unit Relationship

1 Ω = 1 V A^-1 (one volt per ampere)

Rearranging gives the familiar form:

V = IR and I = (V/R) Worked Example: Basic Resistance Calculation

Question: A lamp passes a current of 500 mA when a p.d. of 3.0 V is applied. Calculate its resistance.

Solution Convert current: I = 500 mA = 0.50 A R = (V/I) = (3.0/0.50) = 6.0 Ω

Ohm's Law

Ohm's law states that for a conductor at constant temperature, the current through it is directly proportional to the potential difference across it.

Ohm's Law For a metallic conductor at constant temperature: V ∝ I
This means (V/I) = constant, i.e., resistance is constant. Important Distinction

The equation V = IR is the definition of resistance and applies to all components. Ohm's law is the statement that resistance is constant (only true for ohmic conductors under constant physical conditions).

Components that obey Ohm's law are called ohmic conductors. Their I–V graph is a straight line through the origin.

I–V Characteristics

An I–V characteristic is a graph showing how current varies with potential difference for a component. The shape reveals whether the component is ohmic or non-ohmic.

1. Metallic Conductor at Constant Temperature (Ohmic)

FIG 9.7: I-V Characteristic for an Ohmic Resistor

Show a straight line through the origin with I on the vertical axis and V on the horizontal axis. The line extends into both positive and negative quadrants, demonstrating constant gradient (constant resistance).

Graph shape: Straight line through the origin

Finding Resistance from Graph

For an ohmic conductor, R = (V/I) at any point, OR R = (1/gradient) of the I–V graph (since gradient = (I/V) = (1/R)).

2. Filament Lamp (Non-Ohmic)

FIG 9.8: I-V Characteristic for a Filament Lamp

Show a curve through the origin that is steep near the origin and becomes progressively less steep (flatter) at higher values of V and I. The curve is symmetric in both positive and negative quadrants.

Graph shape: Curve that starts steep and becomes less steep at higher currents

Explanation:

FIG 9.9: Filament Lamp Heating Effect

Show two side-by-side views of a lamp filament: (1) At low temperature - metal ions in a lattice with small vibrations, electrons pass relatively easily. (2) At high temperature - ions vibrate with larger amplitude, causing more frequent collisions with electrons and increased resistance.

Common Misconception

Resistance at a point is NOT the gradient of the tangent - it is the ratio (V/I) at that point (equivalent to the gradient of a line from the origin to that point).

3. Semiconductor Diode

FIG 9.10: I-V Characteristic for a Semiconductor Diode

Show an asymmetric curve with virtually no current for negative V (reverse bias region). For positive V, show a sharp exponential increase in current once V exceeds approximately 0.7 V (forward bias threshold for silicon).

Graph shape: Asymmetric - conducts in one direction only

A diode only allows significant current in one direction, making it useful for rectification.

Resistivity

Resistivity is a property of a material that indicates how strongly it opposes current flow. Unlike resistance, resistivity does not depend on the size or shape of the sample.

Resistivity R = (ρ L/A)
where R = resistance (Ω), ρ = resistivity (Ω m), L = length (m), A = cross-sectional area (m^2)

Rearranging:

ρ = (RA/L)

Understanding the Relationship

FIG 9.11: Effect of Length and Cross-Sectional Area on Resistance

Show two comparisons: (1) Two wires of the same cross-sectional area but lengths L and 2L - doubling length doubles resistance. (2) Two wires of the same length but cross-sectional areas A and 2A - doubling area halves resistance.

Units of Resistivity

From ρ = (RA/L):

Units of ρ = (Ω × m^2/m) = Ω m (ohm metres) Exam Tip

The unit is Ω m (ohm metres), NOT Ω/m or Ω m^-1. This is a common error.

Material Resistivity (Ω m) Classification
Copper 1.7 × 10^-8 Good conductor
Nichrome 1.1 × 10^-6 Conductor (high resistivity)
Silicon 10^3 Semiconductor
Glass 10^12 Insulator
Worked Example: Resistivity Calculation

Question: A tungsten filament is 60 cm long and has diameter 50 μm. Calculate its resistance. (Resistivity of tungsten = 5.3 × 10^-8 Ω m)

Solution

Convert units:

L = 60 cm = 0.60 m Diameter = 50 μm = 50 × 10^-6 m, so radius r = 25 × 10^-6 m

Calculate cross-sectional area:

A = π r^2 = π × (25 × 10^-6)^2 = 1.96 × 10^-9 m^2

Calculate resistance:

R = (ρ L/A) = 5.3 × 10^-8 × 0.601.96 × 10^-9 R = 16 Ω (to 2 s.f.) Worked Example: Finding Resistivity

Question: A wire of length 2.0 m and diameter 0.40 mm has resistance 3.5 Ω. Calculate the resistivity of the material.

Solution Radius = 0.20 mm = 0.20 × 10^-3 m A = π r^2 = π × (0.20 × 10^-3)^2 = 1.26 × 10^-7 m^2 ρ = (RA/L) = 3.5 × 1.26 × 10^-72.0 ρ = 2.2 × 10^-7 Ω m

Light-Dependent Resistor (LDR)

FIG 9.12: LDR Circuit Symbol and Resistance Characteristic

Show the LDR circuit symbol (resistor symbol with arrows pointing inward representing incident light). Include a graph of resistance vs light intensity showing resistance decreasing non-linearly as light intensity increases.

An LDR is a semiconductor device whose resistance changes with light intensity.

LDR Behaviour As light intensity increases, resistance decreases.

Explanation:

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