Resistor Nyquist-Johnson noise formula question

In summary, the Nyquist-Johnson noise formula has $e^{h\nu/kT}-1$ in the denominator because it is derived from the Planck radiation law, which is based on Bose-Einstein statistics.
  • #1
telegraphic
3
0
Hi all -- I had a question about the Nyquist-Johnson noise formula:

[itex]
\[
E_{\nu}^{2}d\nu=\frac{4R_{\nu}hd\nu}{e^{h\nu/kT}-1}\]
[/itex]

which of course can be approximated for most purposes as [itex]$ E_{\nu}^{2}d\nu=4R_{\nu}kTd\nu$[/itex].

I'm just wondering why the formula has [itex]$ exp(h\nu/kT)-1$[/itex] in the denominator? I would have expected [itex]$ exp(h\nu/kT)+1$[/itex] as electrons are fermions, so obey Fermi-Dirac statistics as opposed to Bose-Einstein. Sure I'm missing something obvious!

Cheers
 
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  • #2
The Nyquist-Johnson noise formula is derived from the Planck radiation law, which is based on Bose-Einstein statistics. The denominator, $e^{h\nu/kT}-1$, is related to the occupation probability of the electromagnetic modes (or photons) of frequency $\nu$ at temperature $T$. Since photons are bosons, they obey Bose-Einstein statistics, and the occupation probability has a different form than that of fermions.
 

1. What is the Resistor Nyquist-Johnson noise formula?

The Resistor Nyquist-Johnson noise formula, also known as Johnson-Nyquist noise or thermal noise, is a mathematical equation used to calculate the amount of noise present in an electrical circuit due to the random thermal motion of electrons within a resistor. It is given by the formula V2 = 4kTRΔf, where V is the noise voltage, k is the Boltzmann constant, T is the temperature in Kelvin, R is the resistance of the resistor, and Δf is the bandwidth of the circuit.

2. Why is the Resistor Nyquist-Johnson noise formula important?

The Resistor Nyquist-Johnson noise formula is important because it allows scientists and engineers to accurately predict and calculate the amount of noise present in an electrical circuit. This is crucial in designing and analyzing electronic devices, as excessive noise can affect the performance and accuracy of the circuit.

3. How does temperature affect the Resistor Nyquist-Johnson noise?

The Resistor Nyquist-Johnson noise is directly proportional to temperature, meaning that as the temperature increases, so does the amount of noise present in the circuit. This is because higher temperatures cause the electrons in the resistor to have more kinetic energy, resulting in increased thermal motion and thus, more noise.

4. Can the Resistor Nyquist-Johnson noise be reduced?

While it is not possible to completely eliminate the Resistor Nyquist-Johnson noise, it can be reduced by using lower resistance values, reducing the temperature of the circuit, and decreasing the bandwidth. Additionally, using specialized noise-cancelling techniques and components can also help reduce the impact of this type of noise in a circuit.

5. Is the Resistor Nyquist-Johnson noise formula applicable to all resistors?

Yes, the Resistor Nyquist-Johnson noise formula is applicable to all resistors as it is a fundamental property of resistors. However, its effects may be more noticeable in low resistance values, where the noise voltage is relatively larger compared to the signal voltage. In high resistance values, the impact of this noise may be negligible compared to other sources of noise in the circuit.

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