#### Question

A plate current of 10 mA is obtained when 60 volts are applied across a diode tube. Assuming the Langmuir-Child relation \[i_p \infty V_p^{3/2}\] to hold, find the dynamic resistance r_{p}_{ }in this operating condition.

#### Solution

According to Lamgmuir-Child Law,

the relation between plate current (*i*_{p}) and the plate voltage (*V*_{p}) is given by

\[i_p = C {V_p}^{3/2} ............(1)\]

Differentiating equation (1) with respect *V*_{p}, we get:-

\[\frac{d i_p}{d V_p} = \frac{3}{2}C {V_p}^{1/2} ............(2)\]

Dividing (2) and (1), we get:-

\[\frac{1}{i_p}\frac{d i_p}{d v_p} = \frac{3/2C {V_p}^{1/2}}{C {V_p}^{3/2}}\]

\[ \Rightarrow \frac{1}{i_p} . \frac{d i_p}{d v_p} = \frac{3}{2 V_p}\]

The dynamic resistance is given by:-

\[\frac{d v_p}{d i_p} = \frac{2 V_p}{3 i_p}\]

\[ r_p = \frac{2 V_p}{3 i_p}\]

\[ r_p = \frac{2 \times 60}{3 \times 10 \times {10}^{- 3}}\]

\[ r_p = 4 \times {10}^3 = 4 k\Omega\]