Potential difference

A bulb only lights up when a cell pushes charge around the circuit — and that "push" has a name: potential difference. Without it, the free electrons already present inside a wire would just jostle about randomly, with no net drift and no current. A cell (or any other source) does work on charge to move it from one point to another, and the amount of work done per unit charge moved is exactly what we call potential difference. This article builds that idea from the definition $V = W/q$ up to Ohm's Law, $V = IR$ , which lets you connect potential difference, current, and resistance in a single conductor. Along the way you will see how to read the standard circuit-diagram symbols, work through two fully solved numericals, and clear up the traps that most often cost marks in this topic.

The core insight

Potential difference is the work done per unit charge in moving charge between two points, $V = W/q$ , and for a conductor at constant temperature it is linked to current and resistance by Ohm's Law, $V = IR$ .

The core idea

To make charge flow and do useful work — lighting a bulb, running a motor — some agent must do work on that charge, pushing it against its natural tendency to spread out evenly. Inside a cell, chemical energy does exactly this, building up a "push" between the two terminals even before any current flows.

Potential difference between two points is defined as the work done in moving a unit positive charge from one point to the other:

V = \frac{W}{q}

where $W$ is the work done (joules) and $q$ is the charge moved (coulombs). The SI unit of potential difference is the volt (V), with $1\text{ V} = 1\text{ J/C}$ — a potential difference of 1 volt means 1 joule of work is done moving 1 coulomb of charge between the two points. At this level the formula is used directly; you are not required to derive it from first principles. A cell maintains a potential difference between its terminals, and this is what pushes free electrons through the external circuit, producing current. It helps to picture it like a height difference driving water downhill: no height difference, no flow; here it is an "electrical push," not gravity, that drives the charge.

Resistance ( $R$ ) is the opposition a conductor offers to current flow, measured in ohm (Ω). A long, thin wire generally has more resistance than a short, thick one of the same material — there is simply more material for the charge carriers to push through.

Ohm's Law ties potential difference, current, and resistance together for a conductor kept at constant temperature:

V = IR

where $V$ is the potential difference across the conductor (volts), $I$ is the current through it (amperes), and $R$ is its resistance (ohms). Given any two of these quantities, the third follows directly. A circuit diagram uses standard symbols so that anyone can read it without ambiguity: a long line next to a short line represents a cell, with the long line being the positive terminal; a rectangle or zigzag stands for a resistor; a circle (sometimes with an arrow through it) is an ammeter, which is always connected in series so the full current passes through it; a circle with the letter G inside is a galvanometer; and a break in the wire with a small circle at each end represents a key or switch, open when drawn apart and closed when the two circles are joined. Being able to read and draw these symbols correctly is often tested alongside the numerical problems in this chapter.

Figure

A simple circuit with a cell resistor ammeter and key drawn with standard symbols and current arrow showing the loop

Derivation AApplying Ohm's Law to a single conductor

Given the potential difference $V$ across a conductor and its resistance $R$ , find the current $I$ flowing through it, starting only from the definition of resistance as the ratio of potential difference to current at constant temperature, so that the relationship can then be rearranged to solve for whichever one of $V$ , $I$ , or $R$ is unknown in a numerical.

Step 1. For a conductor kept at constant temperature, experiment shows that the current $I$ flowing through it is directly proportional to the potential difference $V$ applied across its ends: $I \propto V$ .

Step 2. Removing the proportionality by introducing a constant for the conductor gives $V = IR$ , where the constant $R$ is defined as the conductor's resistance — this is the statement of Ohm's Law.

Step 3. Because $R$ is a fixed ratio for a given conductor at constant temperature, the same equation rearranges to give whichever quantity is unknown: $R = \dfrac{V}{I}$ when $V$ and $I$ are known, or $I = \dfrac{V}{R}$ when $V$ and $R$ are known.

Step 4. These three forms of the same law — $V = IR$ , $R = V/I$ , $I = V/R$ — are all you need for single-conductor numericals in this syllabus; no combination of multiple resistors is required at this stage.

Worked example

Question: A charge of 12 C passes between two points in a circuit, and 60 J of work is done in moving this charge. Calculate the potential difference between the two points. This potential difference is then applied across a 15 Ω resistor — find the current through it.

Step 1. Write the definition of potential difference: $V = \dfrac{W}{q}$ .

Step 2. Substitute $W = 60\text{ J}$ and $q = 12\text{ C}$ :

V = \frac{60\text{ J}}{12\text{ C}} = 5\text{ V}

Step 3. Now apply Ohm's Law with this potential difference across a resistor of $R = 15\ \Omega$ : $I = \dfrac{V}{R}$ .

Step 4. Substitute the values:

I = \frac{5\text{ V}}{15\ \Omega} = 0.33\text{ A}

The potential difference between the two points is 5 V, and the current through the 15 Ω resistor connected across it is approximately 0.33 A.

Competitive stretch

If instead the same charge of 12 C flowed for 0.33 A of current, note that current is charge per unit time — so checking $I = q/t$ with the numbers above ( $t \approx 12/0.33 \approx 36\text{ s}$ ) is a quick way to sanity-check that your V, I, R, q, W, and t values are all mutually consistent in a multi-part numerical. This kind of cross-check is worth doing whenever a question gives you more information than strictly needed to find one answer — it catches an arithmetic slip before it costs you the next part of the question.

Common traps

Mixing up potential difference (volts, "the push") with current (amperes, "the flow") — related but not the same physical quantity.
Mishandling the formula $V = W/q$ when work is not given directly in joules or charge not in coulombs — always convert first.
Applying $V = IR$ to a whole circuit with multiple resistors — at this level Ohm's Law is applied only to a single resistor/conductor with known V, I, or R; series and parallel combinations are Class X topics.
Assuming potential difference exists only while current flows — a cell has a potential difference between its terminals even in an open circuit, before any current starts.
Drawing the ammeter in parallel instead of in series, or forgetting that a key/switch is simply a break in the wire that can be opened or closed — small circuit-diagram slips like these are common in ICSE board answers.

Ask the tutor: Want to practise a few more mixed numericals on potential difference, current, and resistance, or see how circuit-diagram symbols are drawn step by step?

Why do we need a cell if charge already exists freely in a wire?

Free electrons in a metal wire are already there, moving randomly in all directions due to thermal energy, but this random motion produces zero net current. A cell does not create charge — it does work on the existing free electrons, pushing them in one preferred direction using energy stored inside it. That directed push is what we measure as potential difference, and it is what turns random electron motion into a steady, useful current.

Does a bigger potential difference always mean a bigger current?

Only if the resistance stays the same. Ohm's Law, $V = IR$ , shows current depends on both potential difference and resistance: $I = V/R$ . Doubling $V$ while keeping $R$ fixed doubles $I$ , but if the resistance also changes — for instance a thinner wire is used — the current will not simply double even though the potential difference did. Always check what is being held constant before comparing.

Why is potential difference sometimes called voltage in numericals?

"Voltage" is simply the everyday, non-technical name for potential difference, because it is measured in volts. Your textbook and answer key may use either word — they mean exactly the same physical quantity, $V = W/q$ , and both plug into $V = IR$ identically. Just be consistent about which symbol, $V$ , you use in a solution.

Potential difference

The core idea

Worked example

Competitive stretch

Common traps

Before this

Where this leads