When photons of energies twice and thrice the work function of a metal are incident on the metal surface one after other, the maximum velocities of the photoelectrons emitted in the two cases are \( V_1 \) and \( V_2 \) respectively. The ratio \( V_1 : V_2 \) is
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In the photoelectric effect, the kinetic energy of the ejected electrons depends on the frequency of the incident photons. If the frequency increases, the kinetic energy and velocity of the electrons increase accordingly.
Step 1: Understanding the Question:
Einstein's photoelectric equation relates incident energy, work function, and maximum kinetic energy (and thus velocity). Step 2: Key Formula or Approach:
Einstein's Equation: \( E = \Phi + K_{max} = \Phi + \frac{1}{2} m v^2 \).
Where \( \Phi \) is the work function. Step 3: Detailed Explanation:
Case 1: Incident energy \( E_1 = 2\Phi \).
\[ 2\Phi = \Phi + \frac{1}{2} m V_1^2 \Rightarrow \Phi = \frac{1}{2} m V_1^2 \]
Case 2: Incident energy \( E_2 = 3\Phi \).
\[ 3\Phi = \Phi + \frac{1}{2} m V_2^2 \Rightarrow 2\Phi = \frac{1}{2} m V_2^2 \]
Dividing the two equations:
\[ \frac{\Phi}{2\Phi} = \frac{\frac{1}{2} m V_1^2}{\frac{1}{2} m V_2^2} \Rightarrow \frac{1}{2} = \left( \frac{V_1}{V_2} \right)^2 \]
Taking the square root:
\[ \frac{V_1}{V_2} = \frac{1}{\sqrt{2}} \] Step 4: Final Answer:
The ratio of velocities is \( 1 : \sqrt{2} \).