Current & Charge
Electric Current
Electric current is the rate of flow of charge. Conventional current flows from positive to negative (opposite to electron flow).
where I is current (amperes), Q is charge (coulombs) and t is time (seconds).
One coulomb is the charge that flows when a current of 1 A flows for 1 s.
Charge Carriers
In metals, charge is carried by free (delocalised) electrons. In electrolytes, charge is carried by ions. The number density of charge carriers determines how well a material conducts.
where n = number density of charge carriers, A = cross-sectional area, v = mean drift velocity, e = charge on each carrier.
Key Facts
- Charge is quantised: the smallest unit is the electron charge e = 1.60 × 10−¹&sup9; C
- Charge is conserved: it cannot be created or destroyed
- In metals, drift velocity is typically very small (mm s−¹), even though the electric field propagates near the speed of light
Potential Difference & Power
Potential Difference (Voltage)
The potential difference across a component is the energy transferred per unit charge as charge flows through it.
1 volt = 1 joule per coulomb.
Electrical Power
Power is the rate of energy transfer. Three equivalent forms:
Worked Example
A 12 V battery delivers 2.5 A to a circuit. Calculate the power dissipated.
P = IV = 2.5 × 12 = 30 W
Exam Tip
Choose the power formula that uses the two quantities you know. If you know V and R but not I, use P = V²/R directly.
Resistance & Resistivity
Resistance & Ohm's Law
Resistance is the opposition to current flow. It is defined as the ratio of p.d. to current.
Ohm's law: For an ohmic conductor at constant temperature, the current through it is directly proportional to the potential difference across it (V = IR gives a straight line through the origin).
Resistivity
Resistivity (ρ) is an intrinsic property of a material that quantifies how strongly it resists current.
where R = resistance, A = cross-sectional area, L = length.
Worked Example
A copper wire has length 2.0 m, cross-sectional area 1.0 × 10−² mm² and resistivity 1.7 × 10−² Ω m. Find its resistance.
Convert area: A = 1.0 × 10−² mm² = 1.0 × 10−² m²
R = ρL/A = (1.7 × 10−² × 2.0) / (1.0 × 10−²) = 3.4 Ω
Superconductivity
A superconductor is a material whose resistivity drops to exactly zero below a critical temperature (Tc). Below Tc, current flows without any energy dissipation.
Applications include: MRI scanners, particle accelerators, power transmission, and maglev trains.
Key Facts
- For metals, resistivity increases with temperature (more lattice vibrations impede electron flow)
- Resistivity depends on the material, not the dimensions of the conductor
- Semiconductors have resistivity between metals and insulators
I–V Characteristics
I–V Graphs for Common Components
The shape of the I–V characteristic tells us whether a component obeys Ohm's law and how its resistance changes.
- Ohmic resistor: straight line through origin. Constant resistance. Obeys Ohm's law.
- Filament lamp: curve that flattens at higher V. Resistance increases as temperature rises.
- Diode: current flows in one direction only, above the threshold voltage (~0.6 V for silicon).
Thermistor & LDR
NTC Thermistor: resistance decreases as temperature increases. Used in temperature sensors.
LDR (Light-Dependent Resistor): resistance decreases as light intensity increases. Used in light sensors and automatic lighting circuits.
Exam Tip
To find the resistance at any point on an I–V graph, use R = V/I (not the gradient). The gradient gives 1/R only for a straight line through the origin.
Series & Parallel Circuits
Kirchhoff's Laws
2nd Law (loop rule): Σε = ΣIR (conservation of energy)
Resistors in Series
In series, the same current flows through each component. The total p.d. is shared.
Resistors in Parallel
In parallel, the same p.d. is across each branch. The total current is shared.
Worked Example
Two resistors of 6 Ω and 3 Ω are connected in parallel. Find the total resistance.
1/R = 1/6 + 1/3 = 1/6 + 2/6 = 3/6 = 1/2
R = 2 Ω
Key Facts
- In series: same current, p.d. shared, total R increases
- In parallel: same p.d., current shared, total R decreases
- The total resistance in parallel is always less than the smallest individual resistor
EMF & Internal Resistance
EMF and Terminal p.d.
The EMF (ε) of a source is the total energy transferred per unit charge. Some energy is dissipated inside the source due to its internal resistance (r).
where Vterminal = IR is the p.d. across the external resistance, and Ir is the “lost volts” across the internal resistance.
Measuring EMF and Internal Resistance
Rearranging ε = IR + Ir gives:
This is in the form y = mx + c. Plot V (y-axis) against I (x-axis):
- y-intercept = ε (EMF)
- gradient = −r (negative internal resistance)
Worked Example
A battery has EMF 9.0 V and internal resistance 0.5 Ω. It is connected to a 4.0 Ω resistor. Find the current and terminal p.d.
I = ε / (R + r) = 9.0 / (4.0 + 0.5) = 9.0 / 4.5 = 2.0 A
Vterminal = IR = 2.0 × 4.0 = 8.0 V
Lost volts = Ir = 2.0 × 0.5 = 1.0 V (check: 8.0 + 1.0 = 9.0 V)
Exam Tip
When a voltmeter reads the p.d. across a battery with no current flowing (open circuit), it reads the EMF. As current increases, terminal p.d. falls because more voltage is lost across the internal resistance.
Potential Dividers
The Potential Divider Formula
A potential divider uses two (or more) resistors in series to produce a fraction of the input voltage.
Worked Example
A 12 V supply is connected across two resistors in series: R1 = 4 kΩ and R2 = 8 kΩ. Find Vout across R2.
Vout = 12 × 8/(4 + 8) = 12 × 8/12 = 8.0 V
Sensor Circuits with Potential Dividers
By replacing one resistor with a sensor (thermistor or LDR), Vout changes automatically with the environment:
- Thermistor in R1 position: as temperature rises, R1 falls, so Vout across R2 rises
- LDR in R1 position: as light intensity increases, R1 falls, so Vout across R2 rises
This output voltage can be used to switch on a circuit (e.g. a fan when it gets hot, or a lamp when it gets dark).
Key Facts
- A potential divider only works properly when the output current is negligible (high-impedance load)
- The output voltage is proportional to R2 as a fraction of the total resistance
- Swapping the position of the sensor and fixed resistor inverts the response
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