Question: 5. A graph of potential difference (V) with respect to the electric current (I) for a semiconductor diode is shown below.
(a) Describe the variation of the electrical resistance of the semiconductor diode from V = −0.5 V to V = 0.8 V.
(b) Sketch a graph of potential difference (V) with respect to the electric current (I) for a filament lamp.
(c) A battery is connected to a filament lamp and a switch (See figure below). The electromotive force of the battery is 12.0 V and its internal electrical resistance is 0.50 W. If the potential difference applied across the filament lamp is 12.0 V, the electrical power consumption is 36.0 W.
(c) A battery is connected to a filament lamp and a switch (See figure below). The electromotive force of the battery is 12.0 V and its internal electrical resistance is 0.50 W. If the potential difference applied across the filament lamp is 12.0 V, the electrical power consumption is 36.0 W.
(i) Determine the electrical resistance of the filament lamp when the potential difference across it is 12.0 V.
(ii) Determine the electrical resistance of the filament lamp when the switch is closed and the electric current through the lamp is 2.8 A.
(d) Explain how the two values of electrical resistance determined in part (c) provide evidence for the shape of the graph that you have sketched in part (b).
Mark scheme:
5 (a) The electrical resistance of the diode is very high or infinite for negative voltages to about 0.4V [1]
The electrical resistance of the diode decreases from 0.4 V onwards [1]
(b) The graph is initially a straight line through the origin, and it changes into a curve at a higher voltage with decreasing gradient. [1]
A similar (inverted) graph is shown in the negative voltage region [1]
(c) (i) R = 122 / 36 = 4.0 Ω [1]
(ii) E = 12 = 2.8 × (R + r) [1]
(R + r) = 4.29 Ω [1]
R = 3.8 Ω [1]
(d) The electrical resistance of the filament lamp increases with an increase of potential difference and electrical current. [1]
Possible answers:
(a) For negative voltages and up to about 0.45V, the electrical resistance is very high (or approaches infinity).
For positive voltages beyond 0.45 V, the electrical resistance decreases gradually.
(b) The graph of potential difference (V) with respect to the electric current (I) for a filament lamp is shown below.
(c) (i) P = VI = V2/R Þ R = V2/P = 122/36 = 4.0 Ω
(ii) Assuming the electrical resistance of the wire is zero,
E = I (R + r) Þ 12 = 2.8 (R + 0.5)
Thus, R = 12/2.8 – 0.5 = 3.79 Ω (or 3.8 Ω)
(d) The electrical resistance of the lamp increases from about 3.8 Ω to 4.0 Ω when the potential difference or electric current increases.
Feynman’s insights or goofs:
In his lectures on physics, Feynman explains that “[t]he voltage-current characteristic of Eq. (14.14) I = I0(e+qDV/kT – 1) is shown in Fig. 14–10. It shows the typical behavior of solid state diodes—such as those used in modern computers. We should remark that Eq. (14.14) is true only for small voltages. For voltages comparable to or larger than the natural internal voltage difference V, other effects come into play and the current no longer obeys the simple equation (Feynman et al., 1966, section 14–5 Rectification at a semiconductor junction).” That is, we should not expect the semiconductor diode to obey the simple equation in the real world especially for a higher potential difference applied. However, physics students should know that there are many different models of the diode.
Below are four simple models of the diode:
1. Model 1 (Ideal diode):
An ideal diode behaves like a perfect conductor (0 W and 0 V) when an applied voltage is forward-biased (“positive” voltage) and behaves like a perfect insulator (¥ W) when the applied voltage is reverse-biased (“negative” voltage). In other words, if the ideal diode is reverse-biased, the electric current through the diode is zero because its electrical resistance is very high (or infinite). On the other hand, this diode conducts at 0 V and allows any electric current passing through it. The current-voltage graph of the ideal diode is shown below.
2. Model 2 (An idealized diode that has a cut-in voltage):
This model includes the concept of cut-in voltage (or knee voltage). When a diode is formed by joining a p-type silicon to an n-type silicon, there are diffusions of electrons into the p-type silicon and holes into the n-type silicon such that it results in a potential hill or cut-in voltage. To be realistic, one may adopt a model of idealized diode that has a cut-in voltage of about 0.7 V. This voltage may vary and it is dependent on materials such as silicon or germanium used. The current-voltage graph of the ideal diode having the cut-in voltage is shown below.
3. Model 3 (An idealized diode that has a cut-in voltage and electrical resistance): This model includes the concept of cut-in voltage and electrical resistance. To be more realistic, one may prefer a model of idealized diode that also has a resistor. However, this diode is not a linear device and its electrical resistance is not constant because the electric current through it does not directly proportional to the applied voltage. The current-voltage graph of the ideal diode that has an electrical resistance is shown below.
4. Model 4 (Shockley diode):
The current-voltage characteristic of Shockley diode is based on the equation: I = I0(e+qDV/kT – 1) where I0 is the saturation current of the diode, q is the electric charge of an electron, k is Boltzmann’s constant, and T is the absolute temperature (in Kelvin). In this model, the saturation current is proportional to the cross-sectional area of the diode. The current-voltage graph of a Shockley diode is shown below.
Importantly, in his lectures on computation, Feynman (1996) elaborates that “[i]n the real world, I(V) cannot just keep on growing exponentially with V; other phenomena will come into play, and the potential difference across the junction will differ from that applied. Note also that the current trickle that exists in the reverse-biased case, catastrophically increases (negatively) at a certain voltage, the so-called breakdown voltage (p. 221).” For instance, a breakdown-induced “meltdown” due to a higher potential difference applied such that the diode does not approximately obey Shockley’s equation. However, physics teachers should also have a deeper understanding how physicists and electrical engineers idealize the current-voltage characteristics of a semiconductor diode under different temperature conditions.
References:
1. Feynman, R. P. (1996). Feynman lectures on computation. Reading, Massachusetts: Addison-Wesley.
2. Feynman, R. P., Leighton, R. B., & Sands, M. (1966). The Feynman lectures on physics Vol III: Quantum Mechanics. Reading, MA: Addison-Wesley.