#### Question

The power delivered in the plate circular of a diode is 1.0 W when the plate voltage is 36 V. Find the power delivered if the plate voltage is increased to 49 V. Assume Langmuir-Child equation to hold.

#### Solution

Given:-

When plate voltage, V_{p}_{, }is 36 V, power delivered in the plate circular of a diode, P, is 1.0 W.

Let the plate current be I_{p}.

Let the power delivered be P' and plate current be I_{p}' when plate voltage, V_{p}, is increased to 49 V

\[ P = I_p V_p \]

\[ \Rightarrow I_p = \frac{P}{V_P} = \frac{1}{36}\]

According to Langmuir-Child equation,

\[I_p \propto ( V_p )^{3/2} , \]

\[ I_p ' \propto ( V_p ' )^{3/2} , \]

\[ \Rightarrow \frac{I_p}{I '_p} = \frac{( V_p )^{3/2}}{( V_p ' )^{3/2}}\]

\[ \Rightarrow \frac{1/36}{I_p '} = \left( \frac{36}{49} \right)^{3/2} \]

\[ \Rightarrow I_p = 0 . 04411\]

Thus, power delivered when the plate voltage is increased to 49 V,

\[P' = V_p ' \times I_p '\]

\[P' = 49 \times 0 . 04411 W\]

\[P' = 2 . 1613 W = 2 . 2 W\]