A recent post in Homework Help had a solution that claimed that the voltage across a diode was 10.5 V. How reasonable/unreasonable is this?
So it got me to pondering the following problem:
If I have a diode that obeys the diode equation ideally (i.e., no equivalent series resistance and no heating effects) that conducts 1 mA of current with 0.7 V across it at room temperature, what is the voltage across it when every electron in the known universe is flowing through it every second?
What is your gut feel, without making any attempt to estimate it at all?
What is your best mental estimate?
What is you best estimate after running some numbers?
Did the results surprise you?
Feel free to post all of your reasoning and work. But try to come up with your answers before reading anyone else's responses. Mine are in the following spoiler.
So it got me to pondering the following problem:
If I have a diode that obeys the diode equation ideally (i.e., no equivalent series resistance and no heating effects) that conducts 1 mA of current with 0.7 V across it at room temperature, what is the voltage across it when every electron in the known universe is flowing through it every second?
What is your gut feel, without making any attempt to estimate it at all?
What is your best mental estimate?
What is you best estimate after running some numbers?
Did the results surprise you?
Feel free to post all of your reasoning and work. But try to come up with your answers before reading anyone else's responses. Mine are in the following spoiler.
My gut feel -- What prompted me to think of the problem in response to the proposed solution is that 10 V struck me as being enough to achieve this. In fact, I wouldn't have been surprised to find that it is a few volts less than this but would not have argued too much if someone claimed that it was a bit more than this. But I would have absolutely balked if someone claimed it was 100 V because I think I have a decent appreciation for the rate of growth associated with exponential relationships.
My best mental estimate -- I know that there are approximately 10^80 protons/neutrons/electrons in the known universe. I also know that at room temperature the current in a diode increases by an order of magnitude for every 60 mV increase in forward voltage. I also know that there are about 10^19 electrons in a coulomb. So assuming that 10^80 is the number of electrons in the universe that would be 10^61 A or 10^64 mA. So 64 orders of magnitude at 60 mV each. That would in turn be about 3.8 V more than the 0.7 V at 1 mA, or about 4.5 V.
Running the numbers -- The best estimate I can find for the number of hydrogen atoms in the observable universe is 10^80, so I can just use that as the number of electrons by assuming that the universe as a whole is neutrally charged. Probably not too far off. With an electronic charge of -1.601e-19 C/e-, the current we are hypothesizing is 1.602e+61 A.
As is almost always done at anything except extremely small currents, I will neglect the "-1" in the diode equation. Given a known voltage, Vo, at a known current, Io, we can find the voltage at other currents with
\(
V_d \; = \; V_0 \; + \; V_t \ln\( \frac{I}{I_0} \)
\)
Vt is the thermal voltage at room temperature, which is 25.85 mV.
Plugging all this in we get.
\(
V_d \; = \; 0.7 \, V \; + \; \(25.85 \, mV \) \ln\( \frac{1.602 \times 10^{61} \, A}{1 \, mA} \)
v_d \; = \; 4.52 \, V
\)
This is a result worth remembering -- it would take less than 5 V to push every electron in the universe through a diode. In fact, 5 V would push all of the electrons in about 100 million universes!
So when people are getting answers that are much more than a volt or so (unless they are taking into account effective series resistance), you can trot out this little factoid.
Did the result surprise me -- I don't think so. I think I would have not been surprised by anything between 5 V and 15 V, so this is slightly below the low side of that range, but not enough to shock me. If anything, it was that last little calculation -- that an extra half a volt to get to an even 5 V, increases it from one universe to a hundred million universes -- that made the biggest impact.
My best mental estimate -- I know that there are approximately 10^80 protons/neutrons/electrons in the known universe. I also know that at room temperature the current in a diode increases by an order of magnitude for every 60 mV increase in forward voltage. I also know that there are about 10^19 electrons in a coulomb. So assuming that 10^80 is the number of electrons in the universe that would be 10^61 A or 10^64 mA. So 64 orders of magnitude at 60 mV each. That would in turn be about 3.8 V more than the 0.7 V at 1 mA, or about 4.5 V.
Running the numbers -- The best estimate I can find for the number of hydrogen atoms in the observable universe is 10^80, so I can just use that as the number of electrons by assuming that the universe as a whole is neutrally charged. Probably not too far off. With an electronic charge of -1.601e-19 C/e-, the current we are hypothesizing is 1.602e+61 A.
As is almost always done at anything except extremely small currents, I will neglect the "-1" in the diode equation. Given a known voltage, Vo, at a known current, Io, we can find the voltage at other currents with
\(
V_d \; = \; V_0 \; + \; V_t \ln\( \frac{I}{I_0} \)
\)
Vt is the thermal voltage at room temperature, which is 25.85 mV.
Plugging all this in we get.
\(
V_d \; = \; 0.7 \, V \; + \; \(25.85 \, mV \) \ln\( \frac{1.602 \times 10^{61} \, A}{1 \, mA} \)
v_d \; = \; 4.52 \, V
\)
This is a result worth remembering -- it would take less than 5 V to push every electron in the universe through a diode. In fact, 5 V would push all of the electrons in about 100 million universes!
So when people are getting answers that are much more than a volt or so (unless they are taking into account effective series resistance), you can trot out this little factoid.
Did the result surprise me -- I don't think so. I think I would have not been surprised by anything between 5 V and 15 V, so this is slightly below the low side of that range, but not enough to shock me. If anything, it was that last little calculation -- that an extra half a volt to get to an even 5 V, increases it from one universe to a hundred million universes -- that made the biggest impact.
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