Electric current as the rate of flow of charge, the equation Q = It, charge carriers and number density, the equation I = nAvq for current, and Kirchhoff's first law as conservation of charge.

What this dot point is asking

AQA specification point 3.5.1.1 wants you to define current as the rate of flow of charge, use $Q = It$, relate current to the number density and drift velocity of charge carriers through $I = nAvq$, and apply Kirchhoff's first law as a statement of the conservation of charge.

Current and charge

The coulomb is therefore a derived unit: $1 \text{ C} = 1 \text{ A s}$, the charge passing in one second when the current is one ampere. Charge is quantised in multiples of the elementary charge $e = 1.6 \times 10^{-19} \text{ C}$, so any measurable charge is $Q = ne$ for an integer number $n$ of elementary charges. This quantisation was first demonstrated by Millikan's oil-drop experiment, which found that the charge on a droplet was always an integer multiple of a fixed smallest value.

Conventional current is defined as the direction in which positive charge would flow, from the positive terminal of a supply round the external circuit to the negative terminal. In a metal the actual carriers are electrons moving the opposite way, but the convention was fixed before the electron was discovered and is kept for consistency.

Charge carriers and drift velocity

The current depends on four things: how many carriers there are per unit volume, how big the conductor is, how fast the carriers drift, and how much charge each one carries.

You can derive this expression by counting carriers. In a time $\Delta t$ the carriers drift a distance $v \Delta t$, so all carriers within a cylinder of length $v \Delta t$ and cross-section $A$ pass a chosen plane. That cylinder has volume $A v \Delta t$ and contains $n A v \Delta t$ carriers, carrying total charge $\Delta Q = n A v \Delta t \times q$. Dividing by $\Delta t$ gives $I = nAvq$.

The expression also explains why different materials carry current so differently. Metals have a very high $n$ (about one free electron per atom), so a tiny drift velocity gives a large current. Semiconductors such as silicon have a much smaller $n$ that rises sharply with temperature, which is why their resistance falls as they get hotter. Insulators have almost no free carriers, so essentially no current flows for ordinary voltages.

Kirchhoff's first law

At a junction where three wires meet, if currents $I_1$ and $I_2$ flow in and $I_3$ flows out, then $I_1 + I_2 = I_3$. In a single series loop with no junctions this reduces to the current being the same at every point.

Try this

Q1. Define electric current. [1 mark]

• Cue. The rate of flow of electric charge.

Q2. A charge of $30 \text{ C}$ passes a point in $12 \text{ s}$. Calculate the current. [1 mark]

• Cue. $I = \dfrac{Q}{t} = \dfrac{30}{12} = 2.5 \text{ A}$.

Q3. State Kirchhoff's first law and the conservation principle it expresses. [2 marks]

• Cue. Total current into a junction equals total current out; conservation of charge.