B.5 Current and circuits

A cell as an energy source

A cell transfers energy from a non-electrical source, such as chemical or solar energy, to charge carriers. The energy source establishes an electromotive force (emf) that can drive charge around a circuit.

Meaning of emf

The emf is the energy supplied by the source per unit charge when charge passes through the source. Its unit is the volt, $$1\,\mathrm V=1\,\mathrm{J\,C^{-1}}$$.

Follow the energy

The cell is not a reservoir of charge that gets used up. Charge circulates; the cell supplies energy that is transferred in circuit components such as lamps, motors and resistors.

Common trap

Emf is not the same as current. Emf is energy per charge supplied by the source; current is charge flow per unit time.

Chemical cells

A chemical cell uses chemical reactions to separate charge and provide energy per unit charge. It can provide electrical output without light, but its reactants are finite and may need replacement or recharging.

Solar cells

A photovoltaic cell converts photon energy directly into electrical energy. Its output depends on illumination and cell area; it does not generate alternating current by itself.

Compare the source, not only the circuit

Both sources provide emf and can drive a circuit. The energy-conversion mechanism, availability of input energy, output variability and storage requirements distinguish them.

Common trap

A solar cell is an energy converter, not a storage device. A secondary cell may store the electrical energy produced.

Resistance

Resistance is the ratio of potential difference across a component to current through it:

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

Its SI unit is the ohm, Ω.

Interpret the ratio

For a given current, a larger potential difference means larger resistance. Resistance describes how strongly a component opposes charge flow under the stated operating conditions.

Unit check

From $$R=V/I$$, $$1\,\Omega=1\,\mathrm{V\,A^{-1}}$$. Use the voltage across the component, not the emf of the whole source unless they are equal in the circuit.

Common trap

Resistance is not the same as current. A component can have high resistance and a small current for a given voltage.

Ohm’s law

At constant temperature, an ohmic conductor has $$V\propto I$$, so $$R=V/I$$ is constant. Its I–V graph is a straight line through the origin when plotted with V and I consistently.

Non-ohmic behaviour

A non-ohmic component has a changing resistance, so current is not directly proportional to potential difference. Filament lamps, diodes and thermistors can be non-ohmic.

Why temperature matters

Heating can change a conductor’s resistance. Apply Ohm’s law only under the stated constant-temperature condition; otherwise the slope or ratio changes as the component operates.

Common trap

A curved I–V graph is not automatically wrong. It is evidence that the component is non-ohmic under those operating conditions.

Electrical power

Power is the rate of electrical energy transfer. For a resistor,

$$P=IV=I^2R=\frac{V^2}{R}$$

Choose the convenient form

Use $$P=IV$$ when current and voltage are given, $$P=I^2R$$ when current and resistance are given, and $$P=V^2/R$$ when voltage and resistance are given.

Interpret the unit

A watt is a joule per second: $$1\,\mathrm W=1\,\mathrm{J\,s^{-1}}$$. In a resistor, the transferred electrical energy becomes mainly internal energy and may produce heating.

Common trap

For alternating-current questions, distinguish peak values from mean or rms values. Use the convention and data supplied by the question.

Energy over time

For a steady direct current,

$$E=Pt=IVt$$

where E is electrical energy transferred in time t.

Build the relation

Potential difference is energy per charge, $$V=E/q$$, and current is charge per time, $$I=q/t$$. Combining them gives $$E=VIt$$.

Units and billing

Use seconds for t to obtain joules. Electricity billing may use kWh: $$1\,\mathrm{kWh}=3.6\times10^6\,\mathrm J$$.

Common trap

Do not use power in place of energy. Power is the rate of transfer; multiply by time for total energy.

Energy conversion in a cell

A chemical cell converts chemical potential energy into electrical energy. Internal chemical processes separate charge and maintain an emf between the terminals.

In a complete circuit

When the circuit is closed, charge flows and energy supplied by the cell is transferred to components. The cell’s chemical energy decreases as electrical energy is delivered.

Source versus load

The cell is the source of energy; a resistor, lamp or motor is a load where electrical energy is transferred to other forms. The charges circulate through both.

Common trap

A cell supplies energy, not a continuous supply of new electrons. The same charge carriers circulate through the circuit.

Conventional current

Conventional current is defined as the direction positive charge would move. In an external circuit it flows from the positive terminal of a source toward the negative terminal.

Do not confuse carrier motion

In a metal, mobile electrons move opposite to the conventional-current direction. Circuit arrows still follow the conventional definition unless the question explicitly asks for electron flow.

Apply it to a diagram

Trace from the positive terminal through the external components and back to the negative terminal. The current direction is consistent through a single series branch.

Common trap

The direction of conventional current is not reversed just because electrons are the actual mobile carriers in a metal.

Current

Electric current is charge passing a point per unit time:

$$I=\frac{\Delta q}{\Delta t}$$

The SI unit is the ampere, $$1\,\mathrm A=1\,\mathrm{C\,s^{-1}}$$.

Find charge or carrier count

$$\Delta q=I\Delta t$$

If each carrier has charge magnitude e, the number of carriers passing is $$N=\Delta q/e$$.

Microscopic picture

Metal electrons move randomly with a small drift superimposed when current flows. Current measures net charge flow, not the total random motion of every electron.

Common trap

Use time in seconds and charge in coulombs. Do not use I/t for charge; current is already charge divided by time.

Direct current (DC)

Direct current flows in one direction. Its charge-carrier drift direction does not periodically reverse; a cell-powered circuit is the standard example.

Alternating current (AC)

Alternating current periodically reverses direction. The potential difference and current can vary with time, so instantaneous and rms values must not be confused.

Read the syllabus boundary

The distinction is required here; detailed AC-circuit analysis is not. If an AC waveform is supplied, identify reversal, period and the value the question asks for.

Common trap

A current can have changing magnitude and still remain DC if its direction does not reverse. Direction, not constancy of size, is the defining distinction.

Junction rule

At a circuit junction, total current entering equals total current leaving:

$$\sum I_{\mathrm{in}}=\sum I_{\mathrm{out}}$$

This is conservation of electric charge.

Loop rule

Around any complete circuit loop, the algebraic sum of potential changes is zero:

$$\sum\Delta V=0$$

This is conservation of energy.

Set signs consistently

Choose current directions and a loop direction before writing equations. A negative solved current means the actual direction is opposite to the chosen arrow.

Common trap

Kirchhoff’s laws are applications of conservation laws: the junction rule is charge conservation; the loop rule is energy conservation.

Resistance of a uniform conductor

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

where ρ is resistivity, L is length and A is cross-sectional area. Resistivity is a material property at the stated conditions.

Read the scaling

At fixed material, doubling length doubles R. Doubling diameter makes area four times larger and reduces R to one quarter. A longer, thinner wire has greater resistance.

Units

Resistivity has SI unit Ω m. Use $$A=\pi r^2$$ for a circular wire and convert radius/diameter to metres before calculating.

Common trap

Do not treat resistivity as the resistance of every sample of a material. Geometry changes resistance even when ρ is unchanged.

Series rules

In series, the same current passes through each component:

$$I=I_1=I_2$$

Potential differences and resistances add: $$V=V_1+V_2$$ and $$R_s=R_1+R_2$$.

Parallel rules

In parallel, each branch has the same potential difference:

$$V=V_1=V_2$$

Currents add at the junction and reciprocal resistances add:

$$I=I_1+I_2,\quad \frac1{R_p}=\frac1{R_1}+\frac1{R_2}$$

Solve systematically

Identify junctions and branches, replace simple groups with equivalent resistance, then use Ohm’s law and conservation rules to recover branch currents and voltage drops.

Common trap

Do not use the series current rule in a parallel branch or add parallel resistances directly.

Real-cell model

A real cell has emf ε and internal resistance r. With external resistance R and current I,

$$\varepsilon=I(R+r)$$

The internal resistance accounts for energy transferred inside the cell.

Terminal potential difference

The terminal voltage across the external load is

$$V=IR=\varepsilon-Ir$$

As current increases, the internal voltage drop Ir increases and terminal voltage falls.

Use a graph

A graph of terminal V against I has intercept ε and gradient −r. A graph of ε against I with total resistance has slope R+r.

Common trap

The emf is not always the same as the terminal voltage. They are equal only when current is zero or internal resistance is negligible.

Variable resistance

A variable resistor lets the resistance in a circuit be changed. Increasing the resistance of a series variable resistor reduces the current for a fixed supply voltage. A rheostat normally uses two terminals to control current; a potentiometer uses three terminals as a potential divider.

Predict the circuit response

For a fixed supply, use $I=V/R_{total}$. If the variable resistance increases, total resistance increases and current decreases. In a series circuit, the potential difference across the variable resistor increases while the potential difference across a fixed series component decreases.

Sensor examples

An LDR has resistance that depends on incident light intensity. An NTC thermistor has lower resistance at higher temperature. These components allow a circuit to respond to its surroundings, but the resistance–stimulus relationship must be obtained from data or a stated model.

Common trap

Do not assume that “more resistance” means a larger current. First decide whether the supply voltage is fixed and whether the component is in series or parallel. For an internal-resistance investigation, changing a variable resistor is useful because it creates multiple $V$-$I$ data points.