Problem structure
(a) 3, Numerical, Wave mechanics (harmonics).
(b) 5, Symbolic, Integration of resistivity/Ohm's Law.
(c) 5, Numerical, Series expansion and time-averaging.
(d) 4, Numerical, Electromagnetism (Ampere's Law).
(e) 3, Qualitative, Proportional reasoning and thermal effects.
Plan your route before writing. Use equations, diagrams, units, and a clear final justification where needed.
A specialized "plasma-acoustic" sensor consists of a thin cylindrical tube of length $L = 0.850\text{ m}$ and cross-sectional area $A = 1.20 \times 10^{-4}\text{ m}^2$. The tube is filled with a partially ionized gas where the number density of free electrons is $n_e = 4.50 \times 10^{16}\text{ m}^{-3}$. A constant potential difference $V = 12.0\text{ V}$ is maintained across the ends of the tube.
A longitudinal sound wave of frequency $f = 440\text{ Hz}$ is launched into the gas, creating a standing wave with a node at each end of the tube. The local density of the gas $\rho(x, t)$ fluctuates, causing the local number density of electrons $n(x, t)$ to vary proportionally: $n(x, t) = n_e [1 + \epsilon \sin(kx) \cos(\omega t)]$, where $\epsilon = 0.020$ is the amplitude of the density perturbation, $k$ is the wavenumber, and $\omega$ is the angular frequency.
(a)
3 marks
Determine the harmonic number $m$ of the standing wave and calculate the speed of sound $c_s$ in this gas.
(b)
5 marks
Show that the total resistance $R(t)$ of the gas column can be expressed in the form $R(t) = \frac{R_0}{1 + \beta \cos(\omega t)}$ and find an expression for the constant $\beta$ in terms of $\epsilon$. You may assume the electron mobility $\mu$ (where drift velocity $v = \mu E$) is constant throughout the gas.
(c)
5 marks
Using the approximation $(1+x)^{-1} \approx 1 - x + x^2$ for small $x$, determine the time-averaged current $\langle I \rangle$ flowing through the tube. The electron mobility is $\mu = 0.450\text{ m}^2\text{V}^{-1}\text{s}^{-1}$.
(d)
4 marks
The fluctuating current $I(t)$ induces a magnetic field. Calculate the maximum magnetic flux density $B_{max}$ at a radial distance $r = 5.00\text{ mm}$ from the center of the tube at the moment the current is at its peak.
(e)
3 marks
If the temperature of the gas increases, the speed of sound $c_s$ increases. Qualitatively describe how the frequency of the $m$-th harmonic and the time-averaged current would change, assuming the number density $n_e$ remains constant.
