Flick a switch and the lamp on the far side of the room lights up in the same instant — as if something shot down the wire at the speed of light. Yet if you could shrink down and ride along with the actual charges doing the carrying, you would be crawling: the electrons in a copper wire drift along more slowly than a snail, a fraction of a millimetre every second. Both facts are true at once, and holding them together is the whole story of this page. What flows in a wire, how much of it flows, why the wire warms up as it does, and why a thin, long, hot wire fights the current harder than a fat, short, cold one — all of it comes from two local quantities: the current density J and the resistivity \rho.

The familiar circuit-class law V = IR is only the bulk, wire-sized summary of something happening at every point inside the metal. Underneath it lives the real Ohm's law, a statement not about a whole resistor but about the material at a single point: \mathbf{J} = \sigma\mathbf{E}. This page builds from that microscopic picture up to the equations you already know, so that V = IR stops being a rule to memorise and becomes something you can derive.

Current: counting charge per second

Start with the quantity you already have a feel for. An electric current is charge in motion, and we measure it by how much charge passes a chosen cross-section of the wire each second:

The unit is the ampere (\text{A}): one ampere is one coulomb of charge sailing past every second, 1\ \text{A} = 1\ \text{C/s}. In a metal the moving charges are conduction electrons — one or two per atom, set loose to roam the whole crystal. (By a historical accident the arrow of "conventional current" points the way positive charge would move, which is opposite to the electrons' actual drift. It makes no difference to anything below.)

But I alone is a blunt instrument. It tells you the total charge crossing a surface, yet says nothing about where in the wire the charge is flowing, or how crowded that flow is. Push the same 5\ \text{A} through a hair-thin wire and through a fat busbar and the two are worlds apart — the thin wire runs hot, the busbar stays cool. To capture that we need a local description.

Current density: the local flow, and it points somewhere

The current density J is the current per unit cross-sectional area — how tightly packed the flow is at a point. For a uniform current spread evenly over an area A,

Crucially, J is a vector, \mathbf{J}: it points in the direction the (positive) charge flows at that point. Current I is then what you get by adding up the flow through a whole surface — the flux of \mathbf{J} across it, I = \int \mathbf{J}\cdot d\mathbf{A}, which for a uniform flow straight down a wire of area A is just I = JA. Think of \mathbf{J} as the "traffic per lane" and I as the "cars per second past the toll booth": the same total flow can be dense-in-a-narrow-road or sparse-in-a-motorway.

The microscopic picture: drift velocity

Zoom in. The conduction electrons are never still — even with the switch off they rattle around at enormous thermal speeds, roughly 10^6\ \text{m/s}, careering in random directions and going nowhere on average. Switch the current on and you superimpose the faintest collective lean on all that chaos: a slow, steady drift velocity v_d in the direction of the push. It is this tiny drift, not the frantic thermal motion, that carries the current.

Count the charge that drifts past a cross-section in a time \Delta t. If there are n carriers per unit volume, each with charge e, moving at drift speed v_d, then everything within a distance v_d\,\Delta t of the surface gets across. That slug of wire has volume A\,v_d\,\Delta t, holds n A v_d\,\Delta t carriers, and therefore \Delta Q = n e A v_d\,\Delta t of charge. Divide by \Delta t:

The number n is what makes metals such good conductors and what keeps the drift so slow. Copper packs about n \approx 8.5\times 10^{28} free electrons into every cubic metre — an astronomical crowd. Because v_d = J/(ne) and n is so vast, even a healthy current only needs the electrons to inch forward. We are about to see just how slowly.

The real Ohm's law: \mathbf{J} = \sigma\mathbf{E}

Now, what sets the drift going and keeps it steady? An electric field \mathbf{E} inside the wire. The field accelerates each electron, but the electron almost immediately smacks into a jiggling ion, scatters, and starts over — again and again, billions of times a second. The result of "accelerate, collide, accelerate, collide" is not runaway speed but a steady average drift, like a marble sinking through honey. Push twice as hard and it drifts twice as fast, so for an ordinary metal the drift — and hence \mathbf{J} — comes out simply proportional to the field:

This — not V = IR — is Ohm's law in its true colours. It talks about a point in a material, using only local quantities and one number, \rho, that belongs to the substance itself. Copper has \rho \approx 1.7\times 10^{-8}\ \Omega\cdot\text{m}; a good insulator like glass is around 10^{12}\ \Omega\cdot\text{m} — a span of twenty powers of ten, the widest range of any physical property. Resistivity is a property of the stuff; resistance is a property of a particular lump of it. Keep those two straight and the rest is bookkeeping.

From point to wire: deriving R = \rho L / A and V = IR

Take a straight wire of length L and uniform cross-section A, carrying current I. The current density is uniform, J = I/A, and the field points straight along the wire with magnitude E = \rho J = \rho I / A. The voltage across the wire is just the field times the length (field integrated along the path):

The whole clump in the brackets depends only on the wire — its material (\rho) and its shape (L, A), not on the current flowing. Give it a name, the resistance R:

Notice what just happened: V = IR was never a fundamental law of nature. It is what you get when you take the genuine law \mathbf{J} = \sigma\mathbf{E}, assume \rho is constant, and add it up over a uniform wire. Materials for which \rho really is constant — so that V stays proportional to I — are called ohmic. Plenty of things are not.

The ohmic straight line

For an ohmic resistor, plot the voltage across it against the current through it and you get a straight line through the origin — and its slope is the resistance. Drag the slider to change R and watch the line tilt: a steeper line is a bigger resistance, because a given current now costs a bigger voltage.

Hot wires resist more: \rho(T)

Resistivity is not fixed — it depends on temperature. In a metal, heating makes the crystal ions jiggle harder, so the drifting electrons collide more often and get through less easily: \rho rises with T. Over a modest range the rise is very nearly linear:

where \rho_0 is the resistivity at a reference temperature T_0 (often 20^\circ\text{C}) and \alpha is the temperature coefficient of resistivity (per ^\circ\text{C}). For copper \alpha \approx 3.9\times 10^{-3}, so a wire warmed by 100^\circ\text{C} resists about 39\% more. Because R = \rho L/A shares the same linear factor, R(T) = R_0[1 + \alpha(T - T_0)] too. Drag the slider to change \alpha and see the slope of the rise change:

Semiconductors do the opposite. In silicon or germanium, heating shakes more charge carriers loose than it hampers, so n soars and \rho falls as the material warms — a negative temperature coefficient. That single sign flip is why a metal and a semiconductor behave so differently, and it is the basis of the thermistor, a component prized precisely because its resistance swings sharply with temperature.

Why wires warm up: power dissipation

Each time a drifting electron is accelerated by the field and then crashes into an ion, it dumps its gained kinetic energy into the lattice as heat. Multiply that over the whole flood of carriers and the wire warms — this is Joule heating. The power delivered to a resistor is charge-flow times the voltage it falls through:

Substituting V = IR gives the three faces of the same law:

The form P = I^2 R carries a famous lesson. The heat lost in a transmission line grows with the square of the current, so halving the current cuts the loss to a quarter. That is exactly why the grid ships power at hundreds of thousands of volts: for a fixed power P = IV, cranking V up lets I come down, and the I^2 R losses in the cables plummet.

Worked examples

Example 1 — how slow is the drift? A copper wire of cross-section A = 1\ \text{mm}^2 = 1\times 10^{-6}\ \text{m}^2 carries I = 1\ \text{A}. Copper has n = 8.5\times 10^{28}\ \text{m}^{-3} and e = 1.6\times 10^{-19}\ \text{C}. From I = n e v_d A,

That is about 0.07\ \text{mm/s} — a single electron would take over three hours to crawl one metre. And yet the current, the field, and the light are all "instant." Slow charges, fast signal.

Example 2 — resistance of that wire. Let the same wire be L = 10\ \text{m} long, with \rho = 1.68\times 10^{-8}\ \Omega\cdot\text{m}. Then

Example 3 — power it dissipates. With I = 1\ \text{A} flowing through that R = 0.168\ \Omega,

a gentle warming. Push I = 5\ \text{A} through the same wire, though, and P = 25 \times 0.168 = 4.2\ \text{W} — twenty-five times the heat for five times the current, the I^2 penalty in action.

Example 4 — resistance when it heats up. Warm the wire from T_0 = 20^\circ\text{C} to T = 80^\circ\text{C}, a rise of \Delta T = 60^\circ\text{C}, with \alpha = 3.9\times 10^{-3}\ ^\circ\text{C}^{-1}:

A 23\% jump in resistance just from a warm afternoon's worth of heating — which is also why a hot filament resists so much more than a cold one at switch-on.

Check drift velocity yourself

The formula v_d = I/(neA) is worth playing with. Edit the current or the wire's thickness below and run it — notice how enormous n is, and how that forces the drift speed down into the fraction-of-a-millimetre-per-second range no matter what you do.