Two streams of photons, possessing energies equal to twice and ten times the work function of metal are incident on the metal surface successively. The value of the ratio of maximum velocities of the photoelectrons emitted in the two respective cases is x :\[3\]. The value of x is ________.

Hint: When a radiation of sufficient energy greater than the work function of the metal is incident on a metal surface, the electrons inside the atoms of metal surface gain energy and eject out of the surface. This is the Einstein theory of radiation.

By planck's law, \[E{\text{ }} = {\text{ }}h\nu \]where 'h' is known as Planck's constant having value \[6.63{\left( {10} \right)^{34}}\]J s, and 'ν' is the frequency of radiation.
According to Einstein photoelectric equation, kinetic energy.\[ = {\text{ }}h\nu {\text{ }}-{\text{ }}\Phi \] where \[\Phi \]is known as work function.

Formula used:
Photo electric work function:
\[E = {E_0} - {\text{ }}\Phi \]

Complete answer:
Combining the two laws, one given by plank and other by Einstein
\[E = {E_0} - {\text{ }}\Phi \]

It is given that, \[E_0\] for the first photon:
\[ = E{0_1} = {\text{ }}2{\text{ }}\Phi \]
(as photon possesses energy equal to twice the work function of metal).

\[E_0\] for second photon
\[ = E{0_2} = {\text{ }}10{\text{ }}\Phi \]
(as second photon possesses energy equal to ten times the work function of metal)

Thus,
\[{E_1} = 2{\text{ }}\Phi {\text{ }}-{\text{ }}\Phi = \Phi \]
\[{E_2} = 10{\text{ }}\Phi {\text{ }}-{\text{ }}\Phi = 9\Phi \]

Kinetic energy also is dependent on velocity and mass by the relation
\[E = \dfrac{1}{2}m{v^2}\]
Where v is the velocity of a photon. Mass is a constant value hence gets cancelled when the ratio of first photon to the second photon is taken.

Therefore,
\[\dfrac{{{E_1}}}{{{E_2}}} = \dfrac{{{V_1}^2}}{{{V_2}^2}}\]
\[\dfrac{1}{9} = \dfrac{{{V_1}^2}}{{{V_2}^2}}\]

The ratio of maximum velocities of the photoelectrons emitted in the two respective cases is x : 3.
Thus,
\[\dfrac{1}{3} = \dfrac{{{V_1}}}{{{V_2}^{}}}\]
\[\dfrac{1}{3} = \dfrac{x}{3}\]
$x = 3$

Thus, the value of x is $3$.

Note: Minimum energy required for emission of electrons from the surface of metal is called the threshold frequency. The speed and number of the emitted electrons will depend on two factors, namely the colour and intensity of the incident radiation.