Circuits and electrical energy

Charge, current and potential difference

Average current is I=ΔQ/Δt: the amount of charge crossing a section per time interval. The ampere corresponds to coulomb per second. In a metallic conductor, electrons drift opposite to conventional current. The latter is defined by the motion positive charges would have. Choosing a reference direction does not claim that we already know the physical direction.

Potential difference relates the change in electric potential energy to charge: ΔU=qΔV. A volt corresponds to joule per coulomb. Current and voltage are different quantities: current is defined through a section, whereas voltage compares two points. Voltage can exist without current in an open branch. The source does not create circulating charge; it uses another form of energy to maintain a potential difference and transfer energy to the circuit.

Resistance and scope of Ohm’s law

In an ohmic resistor under approximately constant conditions, terminal voltage is proportional to current: V=RI. Resistance R is measured in ohms, Ω=V/A. The relationship describes experimental behaviour; it is not a universal law for all components. Diodes, heating filaments and other elements may have a non-linear or temperature-dependent relationship.

For a homogeneous wire, R=ρℓ/A in a uniform model: ρ is resistivity in Ω·m, ℓ length and A cross-sectional area. A longer wire has greater resistance; a larger cross-section allows lower resistance. Ideal connections are treated as having zero resistance, but real wires dissipate energy. The approximation’s validity depends on how much these voltage drops matter compared with those of the components studied. Circuits here are low-voltage models and exercises.

Nodes and conservation of charge

A node joins branches connected by ideal conductors. In a steady state, there is no significant charge accumulation at the node. Therefore, the sum of entering currents equals the sum of leaving currents. This is Kirchhoff’s current law, often written as the algebraic sum of currents equalling zero. Choose a convention, such as positive entering currents, and keep it throughout the equation.

If 0,8 A enters and 0,3 A leaves along one branch, 0,5 A must leave along the remaining branch. Current is not lost in resistors: electrical energy is transformed there, but charge is conserved. In series without branches, current is the same in every component. In parallel, branches share the same two nodes and voltage; currents may differ according to resistance.

Loops, signs and equivalent resistances

In a lumped circuit in steady state, without significant changing magnetic flux through its loops, the sum of potential changes around a closed loop is zero. Crossing a source from its negative to positive terminal adds a rise +V. Crossing a resistor along the reference current includes a drop −RI. Crossing the resistor in the opposite direction gives +RI.

In series, drops add: V=I(R₁+R₂), so equivalent R is their sum. In parallel, I=V/R₁+V/R₂, so 1/equivalent R=1/R₁+1/R₂. The equivalent resistance of positive parallel resistors is smaller than the smallest branch resistance. A negative calculated current does not invalidate the solution: it means its real direction opposes the chosen one. Always check that equations respect the stated directions.

Power, energy and passive convention

Instantaneous electrical power is p=vi when v and i follow the passive convention: reference current enters the terminal marked positive. Then p>0 means the component absorbs energy and p<0 means it supplies energy. In a resistor with R>0, p=Ri²=v²/R≥0. A source powering the circuit has negative power under this convention because current leaves its positive terminal.

For constant direct current, E=PΔt. A watt is joule per second; a kilowatt-hour is energy: 1 kWh=3,6×10⁶ J. Do not write kW/h for energy consumed. In a model without energy accumulation in fields, the powers of all components sum to zero: what sources supply equals what receivers absorb. This balance independently checks the current equations.

Real sources and responsible interpretation

A real source can be modelled by electromotive force ε and internal resistance r in series. When supplying current I, terminal voltage is V=ε−Ir. The chemical power or power from another conversion mechanism is εI; part, I²r, heats the interior and the remainder reaches the external circuit. Electromotive force is measured in volts despite its historical name: it is not force in newtons.

An ideal ammeter is connected in series and has zero resistance; an ideal voltmeter compares two nodes and has infinite resistance. These ideals help analyse how measurement alters a real circuit. Numerical examples use limited low-voltage sources and conceptual diagrams. Mains installations require equipment and qualified personnel; this unit aims to interpret models and balances, without instructions for work on domestic wiring. Start with nodes and directions, move to equations and finish with units and power.

Worked examples, without skipping steps

Follow the reasoning first. Then cover the solution and try to reconstruct it in your own words.

Now try it yourself

There is no timer. Write what you know, choose a relationship and justify each step. Hints and solutions are available whenever you need them.

Mistakes worth avoiding

The resistor consumes current.

Charge is conserved; the resistor transforms energy.

Parallel currents are always equal.

Voltages are equal; currents depend on branch resistances.

Negative power necessarily means an error.

Under the passive convention, it means energy supply.