Current, Resistance and Ohm's Law

Current is simply how much charge flows past a point each second, named just like the current in a river, and resistance is how hard a material makes that flow. Ohm's law says the push you need grows in step with the flow you want, and it's what lets you predict how much current a wire, heater or bulb will draw and how hot it'll get. $V = IR$ looks like one more formula to memorize, but it's just three plain pieces: $V$ (how hard the push is, in volts), $I$ (how much charge flows past each second, in amps) and $R$ (how much the wire fights back, in ohms). The resistance $R = \rho L/A$ is just as simple: $\rho$ (what the wire is made of), $L$ (how long it is) and $A$ (how thick it is). Bear with the derivation below; it mostly comes down to asking why electrons don't just keep speeding up.

A river of charge. A phone charger pushes 2 A down its cable. That's 2 coulombs every second, about $1.25\times10^{19}$ electrons. Yet each electron only creeps along at a fraction of a millimeter per second. So why does the phone respond instantly? Because the wire is already full of charge. Switching on sends the field down the wire at nearly the speed of light, and every electron starts drifting at once, like water in a hose that's already full.

Current is the rate at which charge crosses a surface, $I = dQ/dt$, measured in amperes (1 A = 1 C/s). Its direction is the way positive charge would move. In metals the carriers are electrons, so they actually drift against the current arrow. That's a sign convention we're stuck with. When the flow is spread over a cross-section, the current density $\mathbf{J}$ tells you the current per square meter and its direction at each point. The current through a surface is then the flux of $\mathbf{J}$:

It's the same integral as electric flux and flux integrals, just with flowing charge in place of field lines.

Where $J = nqv_d$ comes from. Suppose there are $n$ carriers per cubic meter, each with charge $q$, all drifting at speed $v_d$. Pick a cross-section of area $A$. In a time $dt$, every carrier within a distance $v_d\,dt$ of it makes it through. Those carriers fill a cylinder of volume $Av_d\,dt$, so $dQ = nqAv_d\,dt$. Divide by $dt$ and you get $I = nqAv_d$, so $J = I/A = nqv_d$.

Why current is proportional to field. Here's the question that makes Ohm's law interesting: a field exerts a force, and force causes acceleration, so why don't the electrons just keep speeding up? A free electron in the wire feels a force $eE$ and does speed up, but about every $\tau \approx 10^{-14}$ s it collides with the vibrating lattice and loses the velocity the field gave it. On average the collisions act like a drag force $mv/\tau$:

You've seen this equation before: it's a linear first-order equation, and it's the same one a skydiver obeys. Just like the skydiver reaches terminal speed, $v$ settles within a few $\tau$ at $v_d = eE\tau/m$. There's an animation further down that shows this sawtooth averaging out. So

Notice what that says: the current density is proportional to the field, $\mathbf{J} = \sigma\mathbf{E}$, with conductivity $\sigma = ne^2\tau/m$. Its reciprocal $\rho = 1/\sigma$ is the resistivity, a property of the material alone: $1.68\times10^{-8}\ \Omega\cdot\text{m}$ for copper, about $10^{-6}$ for nichrome heater wire, $10^{12}$ and up for glass. It also makes sense that heat matters. A hotter lattice vibrates harder, collisions come sooner and $\rho$ rises, roughly as $\rho_0[1 + \alpha(T - T_0)]$ with $\alpha \approx 0.0039$ per °C for copper.

From the point form to $V = IR$. $\mathbf{J} = \sigma\mathbf{E}$ is about one point in the wire. To get something you can measure with a meter, add it up over the whole wire. Put a voltage $V$ across a uniform wire of length $L$ and cross-section $A$. The field inside is uniform and runs along the wire, so $V = EL$ (that's $\mathbf{E} = -\nabla V$ from Field from Potential, along a straight path). Then

Resistance grows with length and shrinks with area, a lot like traffic on a road: a longer road means more chances to get stuck, and more lanes let more cars through. A device with constant $R$ is ohmic. A diode or a lamp filament isn't, because its $V/I$ changes as the current changes.

Power. Moving a charge $dQ$ through a potential drop $V$ releases energy $V\,dQ$ (Electric Potential), so the power is $P = V\,\frac{dQ}{dt} = IV$. With $V = IR$ this becomes $P = I^2R = V^2/R$. That's Joule heating, which warms toasters on purpose and power lines by accident.

A common mistake. With $P = I^2R$ and $P = V^2/R$ side by side, it's tempting to think they disagree, since one grows with $R$ and the other shrinks. They don't. They're the same power, written for different things held fixed: at a fixed voltage a smaller resistor burns more power, while at a fixed current a larger one does. Two smaller slips to watch for: use the radius in $A = \pi r^2$, not the diameter, and keep resistivity (the material) and resistance (the object) apart.

Building codes size wires with $R = \rho L/A$ and $P = I^2R$: a wire too thin for its current heats until its insulation fails, so a $20$ A circuit needs thicker copper than a $15$ A one. Grids transmit at hundreds of kilovolts because line loss grows as $I^2$. And a whole family of sensors works by watching a resistance change: platinum thermometers, strain gauges glued to bridges and aircraft wings, and the hot-wire sensors that measure the air flowing into a car engine.