Electric Currents in Conductors

The free-electron picture

• A charge in an electric field feels a force; if it is free to move it contributes to a current. Free charges exist in nature — for example in the ionosphere — but inside atoms and molecules electrons and nuclei are bound.
• In bulk matter the molecules are packed so closely that in some materials, notably metals, the electrons are no longer attached to individual nuclei and are practically free to move through the whole solid. Such materials are conductors. In others the electrons stay bound and do not accelerate even in a field — insulators.
• In solid conductors the current is carried entirely by the negatively charged electrons against a background of fixed positive ions. In electrolytic solutions both positive and negative ions move.

Without a field, and with one

• With no field applied, the electrons move because of thermal energy. They collide repeatedly with the fixed ions; after a collision an electron emerges with the same speed but in a completely random direction.
• Hence at any instant, on the average, as many electrons travel in one direction as in the opposite direction. The average velocity is zero and there is no net current.
• When a potential difference is applied across the ends, a field $E$ is set up inside. Each electron feels $F = -eE$, opposite to $E$ because its charge is negative.
• This superimposes a small net velocity — the drift velocity $v_d$ — on the random thermal motion, directed opposite to $E$, i.e. towards the positive end. The random motion continues exactly as before; a drift is simply added to it, and that drift is the current.

NCERT's cylinder: why a steady current needs a cell

• Take a conducting cylinder of radius $R$. Attach to its two flat faces two thin dielectric discs carrying $+Q$ and $-Q$. A field is created, directed from the positive charge to the negative charge, and electrons are accelerated towards $+Q$.
• While they move they constitute a current — but they are moving so as to neutralise the charges, so after a very short while the charges are gone, the field disappears and the current stops.
• If instead the ends are continuously supplied with fresh charge to make up for what the electrons neutralise, there is a steady field in the body of the conductor and hence a continuous current. Cells and batteries are exactly such mechanisms.

Resistance against resistivity — the 3-mark comparison

Ideal meters in a circuit

• An ideal ammeter has zero resistance, so inserting it changes nothing; it is always connected in series.
• An ideal voltmeter has infinite resistance, so it draws no current; it is always connected in parallel and simply reads the potential difference across the branch it spans.
• CBSE's worked circuits: two cells of 6 V and 9 V, each of internal resistance $1\ \Omega$, in a single loop with an ideal ammeter and an ideal voltmeter across the 6 V branch. Total loop resistance $= 1 + 1 = 2\ \Omega$ in both cases; the voltmeter reads 6 V in both.
• Cells aiding: net emf $= 9 + 6 = 15\ \text{V}$, so $I = 15/2 = 7.5\ \text{A}$.
• Cells opposing: net emf $= 9 - 6 = 3\ \text{V}$, so $I = 3/2 = 1.5\ \text{A}$.

Standard results

• No field: average electron velocity zero, no current. With a field: drift superposed on random motion.
• Force on an electron in the conductor, $F = -eE$; drift is opposite to $E$.
• A transient current dies out unless a cell keeps the field steady.
• $R$ is of the conductor; $\rho$ is of the material.
• Ideal ammeter: zero resistance, in series. Ideal voltmeter: infinite resistance, in parallel.