Current, Potential Difference, Resistance and Power

Electricity is the flow of charge. Two quantities define a circuit before any resistor is involved.

Electric current is the rate of flow of charge — how much charge passes a point each second: $I = \frac{\Delta Q}{\Delta t} \quad\Longleftrightarrow\quad \Delta Q = I\,\Delta t$ The unit is the ampere (A), where $1\ \text{A}=1\ \text{C s}^{-1}$. By convention, current flows from the positive to the negative terminal — the opposite direction to the actual electron drift. Charge is quantised: it comes in whole multiples of the elementary charge $e=1.6\times10^{-19}\ \text{C}$, so the number of electrons passing in a time is $N=\Delta Q/e$.

Potential difference (p.d.) is the energy transferred per unit charge as charge moves between two points: $V = \frac{W}{Q}$ The unit is the volt (V), where $1\ \text{V}=1\ \text{J C}^{-1}$. So a p.d. of 3 V across a bulb means each coulomb of charge delivers 3 J of energy to the bulb.

e.m.f. versus p.d. — a distinction examiners reward.

• e.m.f. ($\varepsilon$) is the energy a source (battery, cell) gives to each coulomb of charge, converting chemical (or other) energy into electrical energy.
• p.d. is the energy each coulomb delivers to a component, converting electrical energy into heat, light, etc.

They have the same unit and defining form ($\text{energy}/\text{charge}$) but describe energy going into versus out of the electrical store. (The full treatment of e.m.f. and internal resistance is in the e.m.f. subtopic.)

Resistance measures how strongly a component opposes the current for a given p.d.: $R = \frac{V}{I}$ The unit is the ohm (Ω), where $1\ \Omega = 1\ \text{V A}^{-1}$. A large resistance means a small current for a given voltage.

Ohm's law is a special result, not a universal one. It states that for a metallic conductor at constant temperature, the current is directly proportional to the potential difference across it: $V \propto I \quad (\text{so } R \text{ is constant})$ The crucial condition is constant temperature — drop that clause and you drop a mark. A component that obeys Ohm's law is ohmic; its I–V graph is a straight line through the origin. Many components are non-ohmic (filament lamps, diodes, thermistors): for these the resistance is not fixed, and you must compute $R=V/I$ at the particular operating point you are interested in. Note carefully: for a curved I–V graph the resistance at a point is $V/I$ (the coordinates), not the gradient of the curve.

An I–V characteristic is a graph of current against potential difference for a component. Three shapes are examined again and again.

Ohmic resistor (constant temperature). A straight line through the origin — current proportional to p.d., constant resistance (constant gradient). Reversing the p.d. simply reflects the line into the third quadrant.

Filament lamp. An S-shaped curve that is steepest near the origin and bends towards the voltage axis. As the current rises the filament heats up; the metal ions vibrate more vigorously and scatter the conduction electrons more often, so the resistance increases with temperature. The resistance at any point is $R=V/I$ (read the coordinates) and it grows as you move out along the curve.

Semiconductor diode. A one-way valve: in the forward direction almost no current flows until the p.d. reaches a threshold of about 0.6 V (silicon), after which the current rises steeply; in the reverse direction the diode blocks current. It is strongly non-ohmic.

To read a resistance off any of these curves at a chosen point, take $R=V/I$ using the coordinates of that point. For the ohmic line this gives the same value everywhere; for the lamp and diode it changes from point to point — that is exactly what "non-ohmic" means.

Two conservation laws underpin every circuit calculation.

Kirchhoff's first law (junctions). The sum of the currents into a junction equals the sum of the currents out of it, $\sum I_{\text{in}}=\sum I_{\text{out}}$. This is the conservation of charge — charge cannot pile up at a point.

Kirchhoff's second law (loops). Around any closed loop the sum of the e.m.f.s equals the sum of the p.d.s, $\sum \varepsilon=\sum IR$. This is the conservation of energy — each coulomb gains energy from the sources and loses exactly that energy to the components round the loop.

Series resistors — derivation. In series the same current $I$ flows through each resistor, and by Kirchhoff's 2nd law the p.d.s add, $V=V_1+V_2$. Since $V_1=IR_1$ and $V_2=IR_2$: $IR = IR_1 + IR_2 \;\Rightarrow\; \boxed{R = R_1 + R_2 + \dots}$ The total is always larger than the largest single resistor, and the supply p.d. divides in proportion to resistance.

Parallel resistors — derivation. In parallel each resistor has the same p.d. $V$ across it, and by Kirchhoff's 1st law the currents add, $I=I_1+I_2$. Since $I=V/R$, $I_1=V/R_1$, $I_2=V/R_2$: $\frac{V}{R} = \frac{V}{R_1} + \frac{V}{R_2} \;\Rightarrow\; \boxed{\frac{1}{R} = \frac{1}{R_1} + \frac{1}{R_2} + \dots}$ The total is always smaller than the smallest single resistor — and you must invert the reciprocal sum at the end to get $R$.

Power is the rate of energy transfer. Combining $P=W/t$ with $W=VQ$ and $I=Q/t$ gives the central result: $P = VI$ Substituting $V=IR$ or $I=V/R$ gives two more equivalent forms: $P = VI = I^2R = \frac{V^2}{R}$ The unit is the watt (W), where $1\ \text{W}=1\ \text{J s}^{-1}$. Choose the form whose quantities you already know: $P=VI$ when V and I are given; $P=I^2R$ for a known current and resistance; $P=V^2/R$ for a known p.d. and resistance.

Energy transferred over a time $t$ is $W = Pt = VIt$ measured in joules. Over long times, energy is billed in kilowatt-hours (kW h) — the energy used by a 1 kW device in 1 hour: $1\ \text{kW h} = 1000\ \text{W}\times3600\ \text{s} = 3.6\times10^{6}\ \text{J}.$

Where the energy goes. In a resistor the electrical energy is dissipated as heat (the moving electrons collide with the lattice ions). This is why the $P=I^2R$ form is so useful for resistive heating: doubling the current quadruples the power dissipated, which matters for cable sizing and fuse ratings (Core Practical contexts CP7 and CP8 use exactly these power relationships when investigating resistivity and internal resistance).