Limitations of Ohm's Law

The graph plots voltage $V$ on the vertical axis against current $I$ on the horizontal axis, both positive (first quadrant). The origin is at the lower-left corner.

• The solid curve represents the actual behaviour of a good conductor. It starts at the origin, is nearly straight for small $I$, then bends upward (concave-up, super-linear) as $I$ increases. This means that at higher currents, the voltage rises faster than a proportional increase — the conductor deviates from Ohm's law.
• The dashed straight line is the ideal Ohm's law prediction: $V = R I$, with constant resistance $R$. It is the tangent to the solid curve near the origin, so at small currents the conductor obeys Ohm's law. At larger currents, the dashed line lies below the solid curve, showing that the actual voltage is greater than the Ohm's law value for the same current.

Physical idea: For a good conductor, Ohm's law holds only up to a certain current. Beyond that, the conductor's resistance increases (often due to heating, which increases lattice vibrations and electron scattering). The graph illustrates limitation (a) from the textbook: "V ceases to be proportional to I."

Key formula developed with this figure is Ohm's law in its ideal form:

where:

• $V$ = potential difference across the conductor (volts)
• $I$ = current through the conductor (amperes) …

What the Figure Shows

The figure is a four-quadrant I–V characteristic of a diode, with the axes crossing at the origin. The vertical axis (current $I$) is scaled differently for positive and negative values: the positive side is labelled in mA (milliamperes), with a marked value of 1.5 mA near the top, while the negative side is labelled in μA (microamperes). The horizontal axis (voltage $V$) also uses different scales: the positive side extends to about 0.2 V (right), and the negative side goes to –2 V (left). This asymmetry in scales is crucial — it visually emphasises that the diode behaves very differently for forward and reverse voltages.

The Curve and Its Physical Meaning

The curve itself is a stretched-S shape that passes through the origin. In the third quadrant (negative $V$, negative $I$), the curve runs slightly below the horizontal axis with a shallow dip — this is the reverse bias region, where only a tiny leakage current (on the order of μA) flows. As $V$ approaches zero from the negative side, the curve bends gently through the origin (the S-bend). In the first quadrant (positive $V$, positive $I$), the curve rises sharply after about 0.2 V — this is the forward bias region, where current increases dramatically (to mA levels) once the voltage exceeds a threshold (the forward knee).

This behaviour directly illustrates limitation (b) of Ohm’s law: the relation between $V$ and $I$ depends on the sign of $V$. Reversing the voltage does not produce a current of the same magnitude in the opposite direction — a diode allows current to flow easily in one direction (forward bias) but blocks it in the other (reverse bias).

Key Formula and Symbol Explanation

The textbook uses this figure to show that Ohm’s law ($V = IR$) fails for a diode. Instead, the diode’s current–voltage relation is given by the Shockley diode equation (introduced later in Chapter 14):

Where:

• $I$ is the diode current (in amperes, A)
• $I_0$ is the reverse saturation current (a very small constant, typically in μA or nA)
• $V$ is the applied voltage (in volts, V)
• $\eta$ is the ideality factor (usually between 1 and 2 for silicon diodes) …

The graph plots Current $I$ (in mA) on the vertical axis against Voltage $V$ (in V) on the horizontal axis, both in the first quadrant (positive values only). The curve starts at the origin $(0,0)$ and rises steeply, but instead of continuing upward linearly, it reaches a rounded peak, then descends into a valley, and finally rises again toward the right. This creates an N-shaped profile: a hump followed by a dip.

Two dashed vertical guide lines partition the curve into distinct regions:

• The ascending part from the origin to the peak is labelled "Non-linear region" — here $V$ and $I$ are not proportional, but the current still increases with voltage.
• The descending part from the peak down to the valley is labelled "Negative resistance region" — here an increase in voltage causes a decrease in current, which is the opposite of ordinary ohmic behaviour.

The physical idea this figure teaches is that Ohm's law is not universal. Specifically, the graph illustrates type (c) deviation: the relation between $V$ and $I$ is not unique — for a given current $I$, there can be more than one corresponding voltage $V$. The material shown, GaAs (gallium arsenide), exhibits this N-shaped characteristic, which is exploited in devices like tunnel diodes.

The key formula the textbook develops with this figure is the definition of resistance in the context of non-ohmic materials. For any device, the static resistance (or DC resistance) at a point is given by:

where $V$ is the voltage across the device and $I$ is the current through it. However, in the negative resistance region, the dynamic resistance (or differential resistance) is negative:

$r = \frac{dV}{dI} < 0$ …