Question

For photoelectric emission, tungsten requires light of \[2300\mathop A\limits^ \circ \]. If light of \[1800\mathop A\limits^ \circ \] wavelength is incident, then emission:
A. Takes place
B. Doesn’t take place
C. May or may not take place
D. Depends on frequency

Answer 1

Hint: The threshold frequency of the metal is the frequency corresponding to the minimum energy that is required to eject the electron from the surface of metal.

Formula used:
\[K = h\nu - {\phi _0}\]
Here K is the kinetic energy of the emitted electron, h is the planck's constant, c is the speed of light, \[\nu \] is the frequency of the photon and \[\phi \] is the work function of the metal.
\[c = \nu \lambda \]
Here c is the speed of light, \[\nu \] is the frequency and \[\lambda \] is the wavelength of the electromagnetic wave.

Complete step by step solution:
For the minimum condition, the kinetic energy of the ejected electron is zero.
Using the energy formula,
\[{\phi _0} = h{\nu _0}\]
Using the relation between the frequency and the wavelength,
\[{\phi _0} = \dfrac{{hc}}{{{\lambda _0}}}\]
Putting in the formula of the kinetic energy, we get
\[K = \dfrac{{hc}}{\lambda } - \dfrac{{hc}}{{{\lambda _0}}}\]
\[K = hc\left( {\dfrac{1}{\lambda } - \dfrac{1}{{{\lambda _0}}}} \right)\]

For the emission of the electron, the kinetic energy of the ejected electron must be non-zero. If the energy of the photon exceeds the minimum energy needed to eject the electron then the rest of the energy is transferred as kinetic energy of the ejected electrons.
So,
\[K \ge 0\]
\[hc\left( {\dfrac{1}{\lambda } - \dfrac{1}{{{\lambda _0}}}} \right) \ge 0\]
\[\dfrac{1}{\lambda } - \dfrac{1}{{{\lambda _0}}} \ge 0\]
\[\dfrac{1}{\lambda } \ge \dfrac{1}{{{\lambda _0}}}\]
\[\lambda \le {\lambda _0}\]
As per the given data \[{\lambda _0} = 2300\mathop A\limits^ \circ \] and \[\lambda = 1800\mathop A\limits^ \circ \]
Hence, the emission takes place.

Therefore, the correct option is A.

Note: It is important to remember that the maximum kinetic energy of an ejected electron is directly proportional to the frequency of incident radiation and independent to the intensity of incident radiation.