diff --git a/_psets/2.md b/_psets/2.md index fb35dc55be9b2122aac2e293bfa551b5ad324ae2..16b433cab4b1705b9a12ba5b8e2114f05f0d99ad 100644 --- a/_psets/2.md +++ b/_psets/2.md @@ -172,12 +172,37 @@ volts. {:.question} What size capacitor has voltage fluctuations that match the magnitude of this Johnson noise? +Since a farad is equal to a second per ohm, purely dimensional considerations imply that the noise +energy of the capacitor is approximately $$kT/C$$. Indeed this turns out to be exactly correct. The +energy in a capacitor is $$C V^2 / 2$$, so by the equipartition theorem, $$kT/2 = \langle C V^2 / 2 +\rangle$$. This implies that $$\langle V^2 \rangle = k T / C$$. Thus the capacitor that matches the +voltage fluctuations found in part (b) at room temperature has capacitance + +$$ +\begin{align*} +C &= \frac{k T}{\left \langle V^2 \right \rangle} \\ +&= \frac{\num{1.38e-23} \si{J/K} \cdot 300 \si{K}}{\num{3.3e-12} \si{V^2}} \\ +&= \num{1.25e-9} \si{F} +\end{align*} +$$ + ### (c) {:.question} If it is driven by a current source, how large must it be for the RMS shot noise to be equal to 1% of that current? +RMS shot noise is the square root of $$2 q I \Delta f$$. So we want $$\sqrt{2 q \Delta f I} = 0.01 +I$$, which is solved by + +$$ +\begin{align*} +I &= \frac{2q \Delta f}{0.01^2} \\ +&= \frac{2 \cdot \num{1.6e-19} \si{C} \cdot 20 \si{kHz}}{10^{-4}} \\ +&= \num{6.4e-11} A +\end{align*} +$$ + ## 3.4