A clean metal surface with threshold frequency f_0 is illuminated by monochromatic radiati… — JAMB UTME

A clean metal surface with threshold frequency $f_0$ is illuminated by monochromatic radiation of frequency $2f_0$ and light intensity $I$ , causing emission of photoelectrons with maximum kinetic energy $K_1$ and stopping potential $V_1$ . If the source is changed to radiate light of frequency $3f_0$ while its intensity is simultaneously doubled to $2I$ , what are the new maximum kinetic energy $K_2$ and stopping potential $V_2$ in terms of $K_1$ and $V_1$ ?

$K_2 = 2K_1$ and $V_2 = 2V_1$ Answer
B
$K_2 = 4K_1$ and $V_2 = 4V_1$
C
$K_2 = 4K_1$ and $V_2 = 2V_1$
D
$K_2 = 3K_1$ and $V_2 = 3V_1$

Answer

K_2 = 2K_1

and

V_2 = 2V_1

By Einstein's photoelectric law,

K_{\max} = hf - W_0

. For incident frequency

2f_0

and work function

W_0 = hf_0

, the initial kinetic energy is

K_1 = 2hf_0 - hf_0 = hf_0

. For incident frequency

3f_0

, the new kinetic energy is

K_2 = 3hf_0 - hf_0 = 2hf_0 = 2K_1

. Since stopping potential is related by

eV_s = K_{\max}

V_2 = 2V_1

. Doubling the light intensity from

I

2I

doubles the rate of photoelectron emission but does not alter maximum kinetic energy or stopping potential.

Step-by-Step Solution

Apply Einstein's photoelectric equation to the initial condition.

K_1 = h(2f_0) - hf_0 = hf_0

, and

e V_1 = K_1 = hf_0

The maximum kinetic energy is the energy of the incident photon minus the work function of the metal plate.

Apply Einstein's photoelectric equation to the new frequency condition.

K_2 = h(3f_0) - hf_0 = 2hf_0 = 2K_1

, and

e V_2 = K_2 = 2hf_0 = 2e V_1 \implies V_2 = 2V_1

The new photon energy is

3hf_0

, leaving

2hf_0

of excess energy as photoelectron maximum kinetic energy.

Evaluate the effect of doubling light intensity to

2I

Light intensity has zero impact on individual photoelectron kinetic energy or stopping potential; it only increases the emission rate (saturation photocurrent).

Photoelectric emission is a one-to-one photon-electron interaction process where photon frequency dictates individual energy.

Key Concept

Independence of photoelectron kinetic energy and stopping potential from light intensity

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