To answer the question, let's analyze the scenario step by step:
1. Understanding the Requirement:
- To remove an electron from a metal atom, radiation of frequency [tex]\( x \)[/tex] is required. This implies that [tex]\( x \)[/tex] is the threshold frequency necessary to overcome the work function of the metal atom.
2. Condition Given:
- Radiation of frequency [tex]\( v \)[/tex] is used, and it is specified that the electron [tex]\( e^{-} \)[/tex] is not removed. This provides us with a crucial piece of information that the energy provided by the radiation of frequency [tex]\( v \)[/tex] is not sufficient to remove the electron.
3. Energy Relationship:
- The photoelectric effect is described by the equation [tex]\( E = h \cdot v \)[/tex], where [tex]\( E \)[/tex] is the energy of the incoming photon, [tex]\( h \)[/tex] is Planck's constant, and [tex]\( v \)[/tex] is the frequency of the radiation.
- Similarly, the energy required to remove the electron, termed as the work function [tex]\( \phi \)[/tex], is [tex]\( \phi = h \cdot x \)[/tex].
4. Given Condition:
- Since the electron is not removed, the energy provided by the incoming photon ([tex]\( h \cdot v \)[/tex]) is less than the work function ([tex]\( h \cdot x \)[/tex]).
- Mathematically, this can be expressed as [tex]\( h \cdot v < h \cdot x \)[/tex].
5. Simplifying the Inequality:
- By cancelling out [tex]\( h \)[/tex] from both sides (as Planck's constant is a positive constant and cannot be zero), the inequality simplifies to:
[tex]\[ v < x \][/tex]
Therefore, from the explanation above, the correct statement that describes the condition given in the problem is:
(B) [tex]\( v < x \)[/tex]
