Question

Two discharge tubes have identical material structures and the same gas is filled in them. The length of one tube is 10 cm and that of the other tube is 20 cm. Sparking starts in both the tubes when the potential difference between the cathode and the anode is 100 V. If the pressure in the shorter tube is 1.0 mm of mercury, what is the pressure in the longer tube?

Answer 1

According to Paschen's equation, sparking potential (V) is a function of the product of pressure (P) inside the tube and the length (d) of the tube.
$V=f\left(Pd\right)$
As the sparking potential is same in both the tubes, the product of P and d must be equal for both the tubes.
Length of the smaller tube, d_s = 10 cm
Pressure inside the smaller tube, P_s = 1.0 mm of mercury
Let the pressure inside the longer tube, P_l,, be x mm of mercury.
Length of the longer tube, d_l = 20 cm
$$\begin{gathered} P_s{d}_s=P_l{d}_l \\ \frac{{\mathrm{P}}_s}{{\mathrm{P}}_l}=\frac{d_s}{d_l} \\ \frac{10}{20}=\frac{1\mathrm{mm}}{x} \\ \Rightarrow \frac{20}{10}=\frac{1\mathrm{mm}}{x} \\ \Rightarrow x=\frac{1\mathrm{mm}}{2}=0.5\mathrm{mm} \end{gathered}$$
Hence, the pressure in the longer tube is 0.5 mm of Hg.