Question

What is the plate current in a diode valve under the space charge limited operation, when the plate potential is 60 V? In a diode valve, the plate current is 320 mA, when the plate potential is 240 volts:
$\begin{align}
  & a)30mA \\
 & b)20mA \\
 & c)40mA \\
 & d)10mA \\
 & \\
\end{align}$

Answer 1

Hint: As the given question is related to planar diode, child-Langmuir law can be used to find the relation between current and voltage in a planar diode. Space charge is a concept in which excess electric charge is treated as a continuum of charge distributed over a region of space. Using this law, we can find the current at particular voltage.
Formulas used:
$I\alpha {{V}^{\dfrac{3}{2}}}$

Complete answer:
We were given the value of voltage at certain current. Now, we need to find the value of current at some other voltage.
$\dfrac{{{I}_{1}}}{{{I}_{2}}}={{(\dfrac{{{V}_{1}}}{{{V}_{2}}})}^{\dfrac{3}{2}}}$
Substituting the values in the above formula we get,
\[\begin{align}
  & \dfrac{I}{320}={{(\dfrac{60}{240})}^{\dfrac{3}{2}}} \\
 & I=\dfrac{320}{8}=40mA \\
\end{align}\]

Therefore, the correct option is option c.

Additional information:
Child’s law or child-Langmuir law gives us the maximum space-charge -limited current in a planar diode of infinite radius as a function of length between anode and cathode and potential difference between the anode and cathode. It is also called a three-halves-power law. This law can be derived using the Poisson equation directly. This law will govern the space charge limited flow of electron current between two plane parallel electrodes in vacuum when emission velocities of electrons can be neglected. Space charge is nothing but the net electric charge in a given finite volume. If both positive and negative charges are present, space charge is the excess of total positive charge diffused through the volume in question over total negative charge.

Note:
The relation between the voltage and current in the child’s law is different with respect to the ohm's law. In ohm's law, the net current through it is directly proportional to the voltage or potential difference across the circuit. Whereas in this child’s law case, the relation between them is different. So, during the calculation, use the right formula to find out the current at a given voltage.