D.C. circuits

01

Circuit symbols and connections

A circuit diagram shows what is connected to what. Follow the wires and component terminals before deciding which calculation to use.

Current is charge passing per second, measured in amperes. Potential difference is energy transferred per charge between two points, measured in volts. Current is measured through a path; p.d. is measured across two points.

Draw the components

Use the standard symbols below, with lines for connecting wires. Direct current (d.c.) flows in one direction. The worked models here additionally use steady current, whose value remains constant during each measurement.

For a cell, the longer plate is positive and the shorter plate negative. A battery symbol shows repeated cell pairs. Mark the polarity when it matters to the current direction or meter connection.

A variable resistor changes the resistance included in a two-terminal path. Its adjustment arrow is part of the symbol, not a current-direction arrow. A switch opens or closes a path; a fuse is a protective link that melts and breaks the path if the current is sufficiently large for long enough.

An LED is a light-emitting diode. In forward operation, conventional current enters its anode and leaves its cathode, the side marked by the bar. The small arrows pointing outwards represent emitted light. Use an appropriate series current-limiting resistor in a practical LED circuit; different LEDs need different operating conditions.

Identify the connection points

A junction dot marks connected wires. Where wires cross without a connection, draw a clear bridge or gap so the crossing cannot be mistaken for a junction. Do not add a junction simply because two lines pass near each other.

A node is a set of points joined by ideal connecting wire with no component between them. These points have the same potential in the negligible-wire-resistance model. A wire can bend or take a longer route on the page without changing the node it represents.

Series connection

Components share one path, so the current does not split between them. R1 carries the current entering the complete R2/R3 group; it does not necessarily carry the same current as either individual branch.

Parallel connection

Components or groups share both endpoints. R2 and R3 each connect B to C. Sharing just one point is not enough.

To redraw a circuit, first record each component's two endpoints. Keep those endpoints and junctions unchanged as you move the symbols. An extra plain wire joining B directly to C would change this circuit: it would bypass the resistors rather than provide another drawing of the same network.

Place meters for the quantity you need

An ammeter goes in series in the selected path. In a main lead it measures total current; inside one branch it measures that branch's current. Connecting it directly across the source would make a low-resistance bypass, not the intended current measurement.

A voltmeter connects across the two endpoints of the component or group. Across A/C it measures the whole network's p.d.; across B/C it measures the parallel group's p.d. A reading across a pair of series resistors is not automatically the voltage across either one.

We treat the ammeter's resistance as negligible and the voltmeter's current as negligible in the model calculations. Actual instruments approximate these conditions. The measurement examples show how to pair readings for a component or a whole network.

02

Series circuits

Series components carry the same current. Their individual potential differences add to the p.d. across the complete path.

For each resistor, V = IR uses the voltage across that resistor and the current through it. Here V_s means the voltage across the complete external network.

Why the current is the same

In a steady unbranched circuit, charge does not continually pile up at one component. The same amount of charge per second passes each position. A resistor transfers energy; it does not consume part of the current before the next resistor.

Each coulomb undergoes successive energy transfers as it passes the components. The work per coulomb across the whole path is the sum of the work per coulomb in its parts, so the potential differences add.

Add the series resistances

For two resistors carrying the same current, V₁ = IR₁ and V₂ = IR₂. Adding gives V_s = I(R₁ + R₂). The single equivalent resistance must therefore be their sum.

At fixed supply voltage, a larger total resistance means a smaller current. These example values describe a model; practical components must also have suitable ratings and remain close to the assumed temperature.

What an open switch changes

Opening the only path stops the steady current throughout this simple circuit. Each fixed resistor then has V = IR = 0 across it, but the source does not lose its e.m.f. In the ideal open-switch arrangement, the source voltage appears across the gap.

Zero current does not always mean zero voltage. An open gap can have a p.d. across it even though there is no conducting path through it.

03

Parallel circuits

Parallel branches share the same two endpoints, so they have the same potential difference. Their currents add to the total current entering the group.

Follow the connection points, not the shape of the drawing. Two branches are parallel only when each begins and ends at the same pair of nodes.

The entering current divides at a junction and recombines at the other. Charge is conserved, so in steady conditions the total current entering equals the sum of the branch currents.

Find the effective resistance

Each branch current is the common p.d. divided by its own resistance. Adding them gives I_total = V/R₁ + V/R₂ + ... . Since I_total = V/R_parallel, dividing by the common nonzero p.d. gives:

For exactly two resistors, this can be rearranged to R_parallel = R₁R₂/(R₁ + R₂). The general reciprocal relationship also works for three or more branches.

For positive finite branch resistances, the parallel equivalent is smaller than the smallest branch resistance. The added path lets more total current flow at the same p.d.; it does not require all the current to choose only one path.

Open one branch

If these branches are connected directly across an ideal fixed-voltage source, opening one branch leaves the same p.d. across the remaining branches. Their currents stay the same, while total source current falls.

This prediction depends on the connection and supply condition. If another resistor is shared in series with the branches, their p.d. can change when a branch opens. The complete-circuit example shows why.

04

Solving a complete circuit

Reduce a valid group to find the source current, then work back through the original connections to find each component's voltage and current.

Series groups share a path current. Parallel branches share endpoint voltage. R = V/I must refer to the same component or the same complete group.

Keep the same external endpoints

An equivalent resistance replaces a group at its two external terminals. It gives the same total current for the same p.d. across those terminals. It does not mean every original component had that current or the whole group's voltage.

The parallel group has only 6.0 V across it here. Using 12.0 V for either branch would ignore the voltage across R1. Dividing the group's 6.0 V by total current 1.50 A finds the group's 4.0 Ω equivalent, not the resistance of either individual branch.

A branch current can rise while total current falls

Now open only the 12.0 Ω branch. The remaining path is 4.0 Ω in series with 6.0 Ω, so its total resistance is 10.0 Ω.

The new source current is 12.0/10.0 = 1.20 A. All of it passes through the remaining 6.0 Ω branch. That branch current has risen from 1.0 A to 1.20 A, even though source current has fallen from 1.50 A to 1.20 A.

R1 now has a smaller drop, 1.20 x 4.0 = 4.8 V. This leaves 7.2 V across the 6.0 Ω resistor. The changed voltage share explains its larger current. It was never connected directly across the ideal source.

Test the rules with paired measurements

Use a suitable low-voltage cell or battery, switch and appropriately rated resistors. Disconnect the source while changing connections. Use components whose heating is small enough for the intended constant-resistance comparison.

For the supplied 3.00 V model:

• Series: total current is 0.100 A. The 10.0 Ω resistor has 1.00 V across it and the 20.0 Ω resistor 2.00 V. Whole-network resistance is 3.00/0.100 = 30.0 Ω.
• Parallel: currents are 0.300 A through 10.0 Ω and 0.150 A through 20.0 Ω. Total current is 0.450 A, giving whole-network resistance 3.00/0.450 = 6.67 Ω.

These are expected values for the stated model. Record the actual current and terminal p.d. in an investigation. A real battery need not maintain exactly the same terminal voltage when the circuit changes.

1. Choose the measurement. Put an ammeter in a main lead for total current, or in one selected branch for its current. Put the voltmeter across the exact component or group being investigated.
2. Select the ranges and polarity. Use a suitable d.c. range. Conventional current should enter the ammeter's positive terminal; the voltmeter's positive lead goes to the higher-potential point. Check the zero and read an analogue scale without parallax. A nominal 3 V battery can exceed a 3 V range; establish a suitable range before choosing a finer one.
3. Record positions with readings. Label the component or node pair and include units. Compare main current with the sum of branch currents, or complete p.d. with the series drops.
4. Determine resistance from a matched pair. For one component, divide its p.d. by its current. For the whole network, divide the p.d. across A/C by total current entering it.

When moving one meter between positions, readings are taken at different times. Source drift and resistor heating can affect their comparison. Check the supply and allow conditions to settle; do not assume every discrepancy proves a failure of the circuit rule.

Loose contacts or contact resistance can change readings. Meter resistance, finite resolution and changes in temperature are separate limitations. Repeat readings to assess scatter, but correct a wrong connection or range directly. Finer resolution does not by itself guarantee better accuracy.

Brightness is not a calibrated current reading. A lamp's resistance changes as its filament heats. Do not treat a fixed-resistor calculation as an exact prediction for any lamp.