Question

The work function of a metal is \[W\] and \[\lambda \] is the wavelength of the incident radiation. There is no emission of photoelectrons when:
(A) \[\lambda > \dfrac{{hc}}{W}\]
(B) \[\lambda = \dfrac{{hc}}{W}\]
(C) \[\lambda < \dfrac{{hc}}{W}\]
(D) \[\lambda \leqslant \dfrac{{hc}}{W}\]

Answer 1

Hint: We will make the use of the photoelectric effect equation given by Einstein and we will manipulate accordingly to find the necessary required for the condition such that there is no emission of electrons.

 Complete step by step answer:
The photoelectric effect equation of Einstein is \[K.{E_{\max }} = \dfrac{{hc}}{\lambda } - W\]
Where,
\[\lambda \] is wavelength of incident light
\[W\] is a work function of a metal surface.
For emission of photoelectrons,
\[K.{E_{\max }} > 0\]
So,
$
\dfrac{{hc}}{\lambda } - W > 0 \\
\Rightarrow\dfrac{{hc}}{\lambda } > W \\
\Rightarrow\dfrac{1}{\lambda } > \dfrac{W}{{hc}} \\
\therefore\lambda < \dfrac{{hc}}{W} \\
$
Hence, there will be no emission of photoelectrons under the condition \[\lambda > \dfrac{{hc}}{W}\].So, the correct option is (A).

Additional information:
Photoelectric effect: When electromagnetic waves, such as light, touch a material, photoelectric effect is the emission of electrons. Photoelectrons are called electrons released in this way. In condensed matter and solid state and quantum chemistry the phenomenon is investigated to draw knowledge on the properties of atoms, molecules and solids. Effect has been used in light detection and precision timed electron emission electronics devices. Photoelectric effect can be due to energy transfer from continuous light waves to an electron in classic electromagnetic theory. A change in light intensity would change the kinetic energy of the electrons that are released, and sufficiently faint light would slow the emission of the electrons to build up enough energy to exit the material. But both predictions differ from the experimental findings. Rather they demonstrate that only if the light reaches threshold frequency electrons are dislodged.Photoelectrons: In a photoelectric effect, electrons may be flung from a metal surface when light shines on a metal. Often also called photo emissions are these operations, and photoelectrons are considered the electrons that are emitted from the metal.

Note:The kinetic energy of emitted photoelectrons should increase with the light amplitude. If the light frequency is raised, the rate of electron emission should increase, which is proportional to the electric current measured.