Charge, Current & Voltage

The three quantities every circuit ultimately describes — and how they relate through one simple ratio.

Circuit AnalysisElectrical Engineering Year 1Free preview

⏱️ About 14 min

You flip a switch and a room floods with light — but what actually traveled through the wires to make that happen?

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The big idea: Every circuit is a story about charge in motion, pushed by a voltage difference, and those two quantities — current and voltage — are all you need to start analyzing what happens inside any device.

🎯 By the end, you'll be able to

Define electric charge, current, and voltage in SI units and state the relationships between them
Compute current from charge as a function of time using the time derivative
Distinguish conventional current direction from electron flow direction
Calculate the charge transferred through a conductor given current and time

Electric charge: the stuff that moves

Electric charge, denoted Q or q, is a fundamental property of matter. Its SI unit is the coulomb (C). The smallest discrete unit of charge is the magnitude of the electron's charge:

e = 1.602 × 10⁻¹⁹ C

In a conducting wire, charge is carried by billions of free electrons. A single coulomb corresponds to roughly 6.24 × 10¹⁸ electrons — an enormous number, which is why we work with aggregate charge rather than counting individual particles.

\[ i(t) = \frac{dq}{dt} \]

Current i (amperes, A) is the time rate of change of charge q (coulombs, C) passing through a cross-section. One ampere equals one coulomb per second.

Current: charge in motion

When charge moves, we call that flow an electric current. The SI unit is the ampere (A), where 1 A = 1 C/s.

A critical convention: conventional current is defined as the direction positive charge would flow. In metals, the actual carriers are electrons (negative charge), so electron flow is opposite to conventional current. Throughout this course — and throughout all of circuit analysis — we use conventional current.

Conventional current (defined as positive charge flow) points opposite to electron flow in a metal conductor.

Voltage: the push behind the flow

Voltage, denoted V or v, measures the work per unit charge required to move charge between two points. Its SI unit is the volt (V), where 1 V = 1 J/C.

Think of voltage as the electrical analog of elevation difference in a water pipe: a larger voltage means a stronger 'push' driving current through a circuit. Voltage is always measured between two points — it is a difference, not an absolute value. When we write V_ab = 5 V, it means point a is 5 V higher than point b. In practice, we often designate a reference node (ground) at 0 V and express all other node voltages relative to it.

\[ v = \frac{dw}{dq} \]

Voltage v (volts, V) is the work w (joules, J) per unit charge q (coulombs, C) required to move charge between two points.

📝 Worked example: The charge flowing through a wire is given by q(t) = 2t² + 3t C (for t ≥ 0). Find the current at t = 2 s.

Current is the time derivative of charge: i(t) = dq/dt.
Differentiate: i(t) = d/dt(2t² + 3t) = 4t + 3 A.
Substitute t = 2: i(2) = 4(2) + 3 = 8 + 3 = 11 A.

✓ The current at t = 2 s is 11 A.

✏️ Practice: A steady current of 0.5 A flows through a wire. How much charge (in coulombs) passes through a cross-section in 2 minutes?

Solution

Convert time to seconds: 2 minutes = 120 s.
For constant current, Q = I × t = 0.5 A × 120 s = 60 C.

Check your understanding

1. A current of 5 A flows through a wire. How much charge passes through a cross-section in 10 seconds?

Q = I × t = 5 A × 10 s = 50 C. Current is charge per unit time, so multiply current by the time interval.

2. In a metal wire, conventional current flows in which direction relative to electron flow?

Conventional current is defined as the direction positive charge would move. Since electrons carry negative charge, their physical motion is opposite to the conventional current direction. This convention is used universally in circuit analysis.

✅ Key takeaways

Charge (coulombs) is the fundamental quantity; current (amperes) is its rate of flow, i = dq/dt.
Voltage (volts) is work per unit charge, v = dw/dq, and is always measured between two points.
Conventional current is defined as positive charge flow — opposite to electron flow in metals.
One ampere equals one coulomb per second; one volt equals one joule per coulomb.

➡️ Now that charge, current, and voltage are defined, the next step is to combine voltage and current to quantify power and energy — and learn the sign convention that tells you whether a device is absorbing or delivering energy.

Want to test yourself on this? Try the Electrical Aptitude test →