GATE 2022 PH – Question 35
A plasma spectrometer covers $3\times10^{12}$ to $30\times10^{12}$ rad/s. The minimum detectable carrier density is $n\times10^{21}\ \mathrm{m^{-3}}$. Find n to two decimals using $|e|=1.6\times10^{-19}$ C, $m=9.1\times10^{-31}$ kg, $\epsilon_0=8.85\times10^{-12}$ SI.
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Correct answer: 2.82 to 2.84
Explanation
A plasma reflects electromagnetic waves with frequencies below the **plasma frequency**
$$\omega_p^2=\frac{ne^2}{\epsilon_0m}\;\Rightarrow\;n=\frac{\epsilon_0m\,\omega_p^2}{e^2}.$$
The spectrometer covers $3\times10^{12}$ to $30\times10^{12}$ rad/s, so the smallest plasma frequency it can detect is $\omega_p=3\times10^{12}$ rad/s, and the smallest carrier density is
$$n_{min}=\frac{\epsilon_0m\,\omega_p^2}{e^2}.$$
Step by step: $\epsilon_0m=8.85\times10^{-12}\times9.1\times10^{-31}=8.05\times10^{-42}$; times $\omega_p^2=9\times10^{24}$ gives $7.248\times10^{-17}$; dividing by $e^2=2.56\times10^{-38}$:
$$n_{min}=2.83\times10^{21}\text{ m}^{-3},$$
so $n=\mathbf{2.83}$.