The work function of a substance is 4.0 eV. The longest wavelength of light that can cause photoelec

Q: Why does the longest wavelength correspond to the minimum energy?

According to Planck's equation, $E = \frac{hc}{\lambda}$, energy and wavelength are inversely proportional. A longer wavelength means a lower frequency and thus lower energy per photon. The longest wavelength that can cause emission provides exactly the minimum energy needed, which is the work function.

Q: What happens if we use a wavelength longer than $310 \text{ nm}$?

If the wavelength is longer than the threshold wavelength ($310 \text{ nm}$), the energy of the incident photons will be strictly less than the work function ($4.0 \text{ eV}$). Consequently, no photoelectrons will be emitted, regardless of how intense the light is.

Q: Why do we use $12375$ or $12400$ for $hc$?

It is a convenient conversion factor. Since $h = 6.63 \times 10^{-34} \text{ J}\cdot\text{s}$ and $c = 3 \times 10^8 \text{ m/s}$, their product in Joules is cumbersome. Converting it to $\text{eV}\cdot\text{\AA}$ gives approximately $12400$ (or $12375$ for better precision in some cases), which simplifies calculations significantly.

Q: Does the intensity of light affect the threshold wavelength?

No, the threshold wavelength is an intrinsic property of the material (dependent only on its work function) and is completely independent of the intensity of the incident light. Intensity only affects the number of photoelectrons emitted per second, provided the wavelength is below the threshold.

Q: What if the work function was given in Joules?

You would either convert the work function to eV (by dividing by $1.6 \times 10^{-19}$) to use the shortcut formula, or use the standard SI values for $h$ and $c$ to find the wavelength in meters, and then convert to nanometers.

Question

The work function of a substance is $4.0	ext{ eV}$. The longest wavelength of light that can cause photoelectron emission from this substance is approximately

Options:

(A) $540	ext{ nm}$

(B) $400	ext{ nm}$

(C) $310	ext{ nm}$

(D) $220	ext{ nm}$

Sigma Percentile · Accepted Answer

The photoelectric effect is one of the most fascinating phenomena in modern physics, serving as the definitive proof of the particle nature of light. When light shines on a metal surface, it can eject electrons, but only if the light possesses enough energy.

### The Concept of Work Function and Threshold Wavelength

Every metal has a specific energy barrier that an electron must overcome to escape its surface. This minimum energy is known as the **work function** ($W$). In our problem, the work function of the substance is given as $4.0 	ext{ eV}$.

The energy of an incident photon is directly proportional to its frequency and inversely proportional to its wavelength, as given by Planck's equation:
$$ E = \frac{hc}{\lambda} $$

The question asks for the **longest wavelength** of light that can cause photoelectron emission. A longer wavelength corresponds to a lower energy. Therefore, the longest possible wavelength will provide exactly the minimum energy required to eject an electron, leaving it with zero kinetic energy. This specific wavelength is called the **threshold wavelength** ($\lambda_0$).

At the threshold wavelength, the energy of the photon is exactly equal to the work function:
$$ \frac{hc}{\lambda_0} = W $$

### The Magic Shortcut

In competitive exams like JEE, speed is crucial. Plugging in the standard SI values for Planck's constant ($h = 6.63 	imes 10^{-34} 	ext{ J}\cdot	ext{s}$) and the speed of light ($c = 3 	imes 10^8 	ext{ m/s}$) can lead to tedious calculations.

Instead, we use a highly effective shortcut. The product $hc$ can be approximated as:
$$ hc \approx 12400 	ext{ eV}\cdot	ext{\AA} \quad 	ext{or} \quad 12375 	ext{ eV}\cdot	ext{\AA} $$

Using $12375 	ext{ eV}\cdot	ext{\AA}$ (which is often more precise for certain textbook problems), the formula for the threshold wavelength in Angstroms becomes:
$$ \lambda_0 	ext{ (in \AA)} = \frac{12375}{W 	ext{ (in eV)}} $$

### Final Calculation

Now, we simply substitute the given work function $W = 4.0 	ext{ eV}$ into our shortcut formula:
$$ \lambda_0 = \frac{12375}{4.0} 	ext{ \AA} $$

Performing the division:
$$ \lambda_0 = 3093.75 	ext{ \AA} $$

The options provided in the question are in nanometers (nm). We know that $1 	ext{ nm} = 10 	ext{ \AA}$. To convert our result, we divide by 10:
$$ \lambda_0 = \frac{3093.75}{10} 	ext{ nm} = 309.375 	ext{ nm} $$

Looking at the given options, $309.375 	ext{ nm}$ is approximately equal to **$310 	ext{ nm}$**.

Thus, the longest wavelength of light that can cause photoelectron emission from this substance is approximately $310 	ext{ nm}$, making option (c) the correct choice.

Animated Solution for Physics - Dual Nature of Matter and Radiation: The work function of a substance is $4.0 eV$ . The longest wavelength of light that can cause photoelectron emission from this substance is approximately

Select Answer:

Visualized Solution

$Threshold Wavelength$

$Energy-Wavelength Relation$

$Shortcut Formula$

$Substitution$

$Calculation$

$Conversion to nm$

The Sigma Insight: Photoelectric Effect

The Concept of Work Function and Threshold Wavelength

The Magic Shortcut

Final Calculation

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