Conductors, diodes and sensors

Interpret a curve at the operating point

On a graph with current vertically and potential difference horizontally, a constant-temperature ohmic conductor produces a straight line through the origin. Its slope is 1/R, not R. At any nonzero operating point the ratio V/I gives resistance. For a curved relation, the tangent slope dI/dV and the ratio I/V are generally different, so a tangent should not automatically be inverted to obtain the defined static resistance.

A filament lamp’s current increases less rapidly at larger voltage because the filament heats and its resistance rises. Its curve is approximately symmetric under reversing both current and voltage. A diode is asymmetric: little current flows in reverse under ordinary operating conditions, while forward current increases rapidly after the turn-on region. Its exact voltage scale depends on material, temperature and current; a single universal turn-on voltage is not a law for every diode.

A thermistor’s measured I–V curve depends strongly on how temperature is controlled. At fixed bath temperature it may behave approximately resistively over a small low-power range; if current heats it, its resistance changes and the curve can steepen. Build the measurement circuit with a current-limiting resistor, use suitable meter ranges and record component conditions. A graph identifies behaviour under those conditions rather than proving that the component will respond identically in every circuit.

Static resistance at a point: R = V/I; for a straight I-versus-V line through zero: slope = 1/R.

Symbols and units

• V is potential difference in V; I is current in A; R = V/I is static resistance in Ω.

How the equation explains the idea

1. For a straight I-versus-V line through zero, I = V/R and its gradient is 1/R.
2. At a nonlinear operating point, use the coordinate ratio V/I for static resistance. The tangent slope is dI/dV; its reciprocal is differential resistance and generally differs from V/I.

Worked example

A lamp changes resistance

1. At 1.0 V the lamp carries 0.20 A, so R = 5.0 Ω.
2. At 6.0 V it carries 0.60 A, so R = 10 Ω.
3. The larger V/I at the second point is consistent with a hotter filament.

A rising current does not imply a falling resistance when voltage rises as well.

Keep this idea. Read axes and operating conditions, then calculate V/I at the chosen point.

A current from many slowly drifting carriers

Consider a uniform conductor with cross-sectional area A and mobile-carrier number density n. In time Δt, carriers drifting at average speed v move through length vΔt. The number crossing a section is therefore nAvΔt, and each carries charge magnitude q. Dividing the total passing charge by time gives I = nqvA. This geometrical argument assumes a steady uniform drift and one effective carrier type, or treats each type separately.

The drift speed is a small net directed motion superimposed on much faster random microscopic motion. It is not the propagation speed of an electrical signal through a circuit. A lamp can respond quickly when a switch closes because carriers already exist along the circuit and the field change establishes motion; it does not wait for one particular electron to travel all the way from the battery.

Materials differ in both available carrier density and how readily carriers drift under a given electric field. Metals usually have many conduction electrons. Semiconductors can have far fewer carriers until thermal or optical energy releases additional ones; insulators have very few accessible carriers under normal conditions. The current equation organises these factors but does not say that number density alone determines resistivity, because scattering and drift response to field also matter.

I = nqvA; v = I/(nqA); carrier count crossing in Δt = nAvΔt

Symbols and units

• n is carrier number density in m⁻³; q is charge magnitude per carrier in C; v is drift-speed magnitude in m s⁻¹; A is cross-sectional area in m².
• I is current magnitude in A; conventional current is opposite electron drift.

How the equation explains the idea

1. In Δt, carriers within a length vΔt cross the section. Their volume is AvΔt, so their number is nAvΔt and charge magnitude is nqAvΔt.
2. Divide by Δt to get I = nqvA. At fixed I, n and q, doubling diameter quadruples A and reduces drift speed to one quarter.

Worked example

A wire’s slow net drift

1. A wire carries 0.80 A with n = 5.0 × 10²⁸ m⁻³ and A = 1.0 × 10⁻⁶ m².
2. Use q = 1.60 × 10⁻¹⁹ C, giving nqA = 8000 C m⁻¹.
3. Drift speed = 0.80/8000 = 1.0 × 10⁻⁴ m s⁻¹.

This small drift speed does not determine how quickly switching information reaches the lamp.

Keep this idea. Count carriers through a moving volume, and distinguish drift speed from signal propagation.

Why metals, thermistors and LDRs respond differently

In a metal, raising temperature increases lattice vibrations. Conduction electrons are scattered more strongly, reducing their directed drift response for a given electric field. The number of mobile electrons changes relatively little, so the increased scattering generally makes resistance rise. This explains the increasing operating resistance of a heated filament without claiming that electrons are consumed or that the wire necessarily changes length enough to account for the effect.

An NTC thermistor is a semiconductor whose carrier population increases strongly with temperature in its working range. Thermal energy makes more charge carriers available for conduction. Although scattering effects also exist, the carrier increase can dominate, so resistance falls as temperature rises. This response is nonlinear and depends on the thermistor material; do not extrapolate one convenient resistance–temperature pair over an unlimited temperature range.

In an LDR, absorbed light can release extra conduction carriers, so increased illumination usually reduces resistance over the intended range. Distinguish this from heating the component: a carefully controlled experiment holds temperature approximately steady when testing illumination. Sensor response may depend on light spectrum, geometry and settling time, so record more than the word bright. Choosing a sensor circuit requires its measured response and intended conditions, not only a qualitative arrow on a diagram.

Metal: temperature ↑ → lattice scattering ↑ → resistance usually ↑. NTC: temperature ↑ → carriers ↑ → resistance ↓. LDR: illumination ↑ → carriers ↑ → resistance ↓.

Worked example

Two sensors under the same supply

1. An NTC has resistance 4.0 kΩ at one temperature and 1.0 kΩ at a higher temperature.
2. At a low measuring voltage of 2.0 V, currents would be 0.50 mA and 2.0 mA.
3. The comparison assumes the measuring power is low enough not to change the intended temperatures appreciably.

The larger current is consistent with reduced thermistor resistance.

Keep this idea. Identify whether scattering or the available carrier population dominates the material’s response.