Why Is Current the Same in a Series Circuit?

In a steady series circuit, the current is the same through every component because the components share one unbranched path. Charge cannot permanently disappear or pile up at one point: whatever charge passes one component each second must pass the next each second.

That does not mean the voltage is the same everywhere. A voltage drop is a change in energy per unit charge, while current measures charge flow per unit time. Resistors can have different voltage drops even though the same current passes through them.

Current measures charge flow

Electric current is defined as the rate at which charge passes a point. The average current over a time interval is

$$I_{\text{avg}}=\frac{\Delta Q}{\Delta t}$$

For the current at an instant, the definition is $I=\frac{dQ}{dt}$. Here, $\Delta Q$ is the amount of charge that passes during the interval $\Delta t$. One ampere means one coulomb of charge passes a point each second.

Imagine tracing a single wire through a battery and two resistors connected in series. In steady operation, each coulomb that passes the first resistor must continue along the path and pass the second. If fewer coulombs per second passed the second resistor, charge would accumulate in the wire between the resistors. If more passed the second, charge there would drain away. Such changes can happen briefly while a circuit is switched on, but they do not continue in a steady state.

Charge conservation explains the rule

The general rule behind this reasoning is conservation of charge. At a circuit junction, the total current flowing in must equal the total current flowing out, provided charge is not building up at the junction. This is Kirchhoff’s current law.

For an unbranched series path, there is only one incoming and one outgoing route at each point. Therefore, the current entering and leaving each component is equal. If the path branches, the current can divide between branches, but the branch currents add up to the incoming current.

The same idea applies to a series circuit containing components other than resistors. For example, during a capacitor’s charging transient, the current through each component in the single series path is equal at a given instant in the usual lumped-circuit model. After a capacitor has fully charged in a DC circuit, the steady current may become zero everywhere in that path.

Worked example: two series resistors

Connect a $6\,\text{V}$ ideal battery to a $2\,\Omega$ resistor and a $4\,\Omega$ resistor in series. The total resistance is

$$R_{\text{total}}=2\,\Omega+4\,\Omega=6\,\Omega.$$

Using Ohm’s law for the whole circuit gives

$$I=\frac{V}{R_{\text{total}}} =\frac{6\,\text{V}}{6\,\Omega} =1\,\text{A}.$$

So the current through each resistor is $1\,\text{A}$. The voltage drops are not equal:

$$V_1=IR_1=(1\,\text{A})(2\,\Omega)=2\,\text{V},$$ $$V_2=IR_2=(1\,\text{A})(4\,\Omega)=4\,\text{V}.$$

The drops add to the battery voltage: $2\,\text{V}+4\,\text{V}=6\,\text{V}$. The larger resistor has the larger voltage drop, even though the current is the same in both resistors.

Why current can differ in parallel branches

In a parallel circuit, a junction gives charge more than one route. The total current entering the junction divides among the paths. For example, if $3\,\text{A}$ enters a junction and $1\,\text{A}$ flows through one branch, the other branches together carry $2\,\text{A}$ out of the junction.

The current in each parallel branch depends on the voltage across it and its resistance. The current is generally not the same in different branches, but the total entering and leaving a junction remains equal in steady operation.

Common mistakes

• Thinking a resistor uses up current. A resistor transfers electrical energy into other forms, commonly heat, but it does not consume charge. In a steady series path, the current before and after it is equal.
• Assuming equal current means equal voltage. Series components share current, not necessarily voltage. For resistors, $V=IR$, so different resistances at the same current produce different voltage drops.
• Confusing current with charge speed. Current is charge per unit time, not the speed of an individual charge carrier. Carrier drift speed can differ in wires with different cross-sectional areas while the same current passes through them.
• Applying the series rule to separate branches. Current is equal along one unbranched path. At a split, it divides, and the branch currents must be added to find the total current.