Physics

Physics · electromagnetism

Problem

(a) Explain why no Johnson-Nyquist noise is produced by ideal inductors or capacitors. There are multiple explanations; any explanation will be accepted. (Hint: the impedance of an ideal inductor or capacitor is purely imaginary.) Given constants: 2. Johnson-Nyquist noise In this problem we study thermal noise in electrical circuits. The goal is to derive the JohnsonNyquist spectral (per-frequency,

f

) density of noise produced by a resistor,

R

\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R \tag{2}

Here, \langle\rangle denotes an average over time, so

\left\langle V^{2}\right\rangle

is the mean-square value of the voltage fluctuations due to thermal noise.

f

is the angular frequency,

k

is Boltzmann's constant, and

T

is temperature. It says that every frequency range

[f, f+d f]

contributes a roughly equal amount of noise to the total noise in the resistor; this is called white noise. Electromagnetic modes in a resistor We first establish the properties of thermally excited electromagnetic modes

V_{n}(x)=V_{0} \cos \left(k_{n} x-\omega_{n} t\right) \tag{3}

in a resistor of length

L

. The speed of light

c^{\prime}=\omega_{n} / k_{n}

in the resistor is independent of

n

. Context question: (a) The electromagnetic modes travel through the ends,

x=0

and

x=L

, of the resistor. Show that the wavevectors corresponding to periodic waves on the interval

[0, L]

are

k_{n}=\frac{2 \pi n}{L}

. Then, show that the number of states per angular frequency is

\frac{d n}{d \omega_{n}}=\frac{L}{2 \pi c^{\prime}}

. Context answer: 证明题 Context question: (b) Each mode

n

in the resistor can be thought of as a species of particle, called a bosonic collective mode. This particle obeys Bose-Einstein statistics: the average number of particles

\left\langle N_{n}\right\rangle

in the mode

n

\left\langle N_{n}\right\rangle=\frac{1}{\exp \frac{\hbar \omega_{n}}{k T}-1} \tag{4}

In the low-energy limit

\hbar \omega_{n} \ll k T

, show that

\left\langle N_{n}\right\rangle \approx \frac{k T}{\hbar \omega_{n}} \tag{5}

You can use the Taylor expansion

e^{x} \approx 1+x

for small

x

. Context answer: 证明题 Context question: (c) By analogy to the photon, explain why the energy of each particle in the mode

n

\hbar \omega_{n}

. Context answer: 证明题 Context question: (d) Using parts (a), (b), and (c), show that the average power delivered to the resistor (or produced by the resistor) per frequency interval is

P[f, f+d f] \approx k T d f . \tag{6}

Here,

f=\omega / 2 \pi

is the frequency.

P[f, f+d f]

is known as the available noise power of the resistor. (Hint: Power is delivered to the resistor when particles enter at

x=0

, with speed

c^{\prime}

, and produced by the resistor when they exit at

x=L

.) Context answer: 证明题 Extra Supplementary Reading Materials: Nyquist equivalent noisy voltage source The formula

\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R

is the per-frequency, mean-squared value of an equivalent noisy voltage source,

V

, which would dissipate the available noise power,

\frac{d P}{d f}=k T

, from the resistor

R

into a second resistor

r

. Context question: (a) Assume that resistors

R

and

r

are in series with a voltage

V . R

and

V

are fixed, but

r

can vary. Show the maximum power dissipation across

r

P_{\max }=\frac{V^{2}}{4 R} . \tag{7}

Give the optimal value of

r

in terms of

R

and

V

. Context answer: 证明题 Context question: (b) If the average power per frequency interval delivered to the resistor

r

\frac{d\left\langle P_{\max }\right\rangle}{d f}=

\frac{d E}{d f}=k T

, show that

\frac{d\left\langle V^{2}\right\rangle}{d f}=4 k T R

. Context answer: 证明题 Extra Supplementary Reading Materials: Other circuit elements We derived the Johnson-Nyquist noise due to a resistor,

R

. It turns out the equation

\frac{d\left\langle V^{2}\right\rangle}{d f}=

4 k T R

is not generalizable to inductors or capacitors.

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