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QPages 352-354: Q10: HC Verma: Concepts of Physics - Volume 2 Electric Current through Gases Pages 352-354: Q10 Solution: A plate current of 10 mA is obtained when 60 volts are applied…

⚠️ Common Exam Traps & Conceptual Pitfalls

When differentiating the Langmuir–Child law $i=C,V^{3/2}$, the incremental resistance is $r= rac{dV}{dI}= rac{2}{3} rac{V}{I}$, not simply $V/I$. Forgetting the $2/3$ factor gives a 50 % over‑estimate of the dynamic resistance.

A plate current of 10 mA is obtained when 60 volts are applied across a diode tube. Assuming the Langmuir-Child relation ip∝Vp3/2i_p \propto V_p^{3/2} to hold, find the dynamic resistance rpr_p in this operating condition.

Step-by-Step Verified Solution

**Answer:** Dynamic resistance rp=4 kΩr_p = 4 \text{ k}\Omega. **Step-by-Step Explanation:** Langmuir-Child law: ip=CVp3/2i_p = C V_p^{3/2}. Differentiating: dipdVp=32CVp1/2\frac{di_p}{dV_p} = \frac{3}{2} C V_p^{1/2}. Dividing: dipip=32dVpVp⇒dVpdip=23Vpip\frac{di_p}{i_p} = \frac{3}{2} \frac{dV_p}{V_p} \Rightarrow \frac{dV_p}{di_p} = \frac{2}{3} \frac{V_p}{i_p}. rp=23×6010×10−3=4000Ω=4 kΩr_p = \frac{2}{3} \times \frac{60}{10 \times 10^{-3}} = 4000 \Omega = 4 \text{ k}\Omega .

All Questions in this Chapter

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