Quick questions on Current and charge: I = nAvq, potential difference and power - Edexcel A-Level Physics

By convention, current direction is the direction of flow of positive charge. In a metal the carriers are actually electrons drifting the opposite way, so conventional current and electron flow point in opposite directions. The total charge that has flowed in a time $t$ is the area under a current-time graph, $Q = \int I \, dt$, which for a steady current is simply $Q = It$.

Consider a conductor of cross-sectional area $A$ containing $n$ free charge carriers per unit volume, each of charge $q$, drifting with mean speed $v$. In a time $\Delta t$ every carrier moves a distance $v \Delta t$, so all carriers within a cylinder of volume $A v \Delta t$ pass a chosen cross-section. The number of carriers is $n A v \Delta t$ and the charge they carry is $\Delta Q = n A v q \, \Delta t$. Dividing by $\Delta t$:

A $3$ kW kettle on a $230$ V UK mains supply draws $I = P/V = 3000/230 = 13$ A, which is why kettle plugs use a $13$ A fuse. The same current in the thin element wire produces a high drift speed and rapid heating, while in the thicker supply cable the larger area keeps the heating per metre low.

In a copper interconnect on a chip with cross-section of order $10^{-13}$ m$^2$, even a microamp gives a drift velocity comparable to a macroscopic wire because $v \propto I/A$. Designers limit current density $J = I/A = nvq$ to avoid electromigration, where fast-drifting electrons physically displace metal atoms and break the track.

Define electric current. [1 mark]

A $12$ V battery delivers $0.50$ A to a lamp for $2.0$ minutes. Find the energy transferred. [2 marks]

Explain why the drift velocity of electrons in a connecting wire is very small even though the lamp lights almost instantly. [3 marks]