Ohm's Law and I-V Curves

Picture a resistor as a pipe full of obstacles that charge has to push through. Voltage is what drives the charge; resistance is how much the material fights back. Double the push and, for a fixed obstacle course, you double the flow. That proportionality is Ohm's law: V = IR. R stays constant because it depends on the material and geometry of the resistor, not on how much current happens to be flowing through it at the moment.

Plot current on the vertical axis and voltage on the horizontal axis, and an ohmic resistor traces a straight line through the origin. The slope of that line is I/V, which equals 1/R. A steep slope means current rises fast per volt, so resistance is low; a shallow slope means resistance is high. The line being straight is the real content of Ohm's law: it says R does not change as V and I change together.

Not everything behaves this way. A diode's I-V curve bends sharply, conducting almost nothing until a threshold voltage and then shooting up. A filament bulb starts as a straight ohmic line at low current, but as current grows it heats the filament, raising its resistance, so the curve bends over and current grows more slowly than voltage. Ohm's law is really a statement about materials, true for conductors held at constant temperature, not a geometric necessity like Gauss's law, which follows from the structure of the electric field itself and never fails. This distinction matters going into power and energy dissipation, P = IV = I^2R, where a changing R changes how a device heats up under load.