Electric Current

What the definition has to contain

• Charges in motion constitute an electric current. Hold a small area normal to the direction of flow; in a time $t$ let $q_+$ be the net positive charge and $q_-$ the net negative charge crossing it in the forward direction. The net charge crossing forward is $q = q_+ - q_-$.
• For a steady current this $q$ is proportional to $t$, and the current across that area in the forward direction is

$$I = \dfrac{q}{t}$$

• A negative value of $I$ simply means the current is in the backward direction.
• Currents are not always steady, so more generally, if $\Delta Q$ crosses a cross-section between $t$ and $t + \Delta t$,

$$I(t) = \lim_{\Delta t \to 0} \dfrac{\Delta Q}{\Delta t}$$

• SI unit: the ampere (A), one of the seven SI base units, defined through the magnetic effect of currents (Chapter 4). $1\ \text{A} = 1\ \text{C s}^{-1}$.

Direction, and why current is still a scalar

• Conventional current is taken along the direction in which positive charge flows. In a metal the carriers are electrons, so the electrons drift opposite to the conventional current — towards the point of higher potential.
• Current is drawn with an arrow but it is a scalar: currents meeting at a junction add algebraically, not by the parallelogram law. Formally $I = \mathbf{j}\cdot\Delta\mathbf{S}$, the scalar product of the current density and the area vector, and a scalar product of two vectors is a scalar.
• Current density $\mathbf{j}$, by contrast, is a vector, directed along $\mathbf{E}$, with magnitude $j = I/A$ and unit A m$^{-2}$.

Orders of magnitude worth quoting

Charge from a graph, electrons from a charge

• When the question supplies an $I$–$t$ plot rather than a number, the charge transferred is the area under the graph.
• CBSE's own worked case: $I$ rises linearly from $0$ to $5\ \text{A}$ between $t=0$ and $t=5\ \text{s}$, then stays at $5\ \text{A}$ to $t=10\ \text{s}$. Then

$$Q = \dfrac{1}{2}\times 5 \times 5 + (10-5)\times 5 = 12.5 + 25 = 37.5\ \text{C}$$

• Number of electrons crossing a section: $N = q/e = It/e$. For $V = 200\ \text{V}$ across $R = 100\ \Omega$ for $1\ \text{s}$: $I = 2\ \text{A}$, $q = 2\ \text{C}$, $N = 2/(1.6\times10^{-19}) = 1.25 \times 10^{19}$.

Standard results

• $I = q/t$ (steady), $I = \mathrm{d}q/\mathrm{d}t$ (general).
• $q =$ area under the $I$–$t$ graph.
• $N = q/e$, with $e = 1.6\times 10^{-19}\ \text{C}$.
• $j = I/A$ in A m$^{-2}$; $I = \mathbf{j}\cdot\Delta\mathbf{S}$.
• In the steady state the same current crosses every section of a circuit element, however its area varies.