Problem 38 - Master Resource Book in JEE Main Physics

Question

The work function of tungsten and sodium are \(4.5 \mathrm{eV}\) and \(2.3 \mathrm{eV}\) respectively. If the threshold wavelength, \(\lambda\) for sodium is \(5460 \AA\), the value of \(\lambda\) for tungsten is (a) \(2791 \dot{\mathrm{A}}\) (b) \(3260 \dot{A}\) (c) \(1925 \mathrm{~A}\) (d) \(1000 \mathrm{~A}\)

Step-by-Step Solution

Answer

The threshold wavelength for tungsten is \(2791 \mathrm{\AA}\), which is option (a).

Step 1: Understanding Work Function and Threshold Wavelength

The work function is the minimum energy required to remove an electron from the surface of a material. The threshold wavelength is the longest wavelength of light that can cause the photoelectric effect for that material. These two are related by the formula: \( \phi = \frac{hc}{\lambda} \), where \( \phi \) is the work function, \( h \) is Planck’s constant \( 4.1357 \times 10^{-15} \mathrm{eV\cdot s} \), \( c \) is the speed of light \(3 \times 10^8 \mathrm{m/s}\), and \( \lambda \) is the threshold wavelength.

Step 2: Relating Work Function and Threshold Wavelength for Sodium

Given: Work function \( \phi_{\text{Na}} = 2.3 \mathrm{eV} \) and threshold wavelength \( \lambda_{\text{Na}} = 5460 \mathrm{\AA} = 5460 \times 10^{-10} \mathrm{m} \). Use the formula \( \phi = \frac{hc}{\lambda} \) to check consistency: \( 2.3 \mathrm{eV} = \frac{(4.1357 \times 10^{-15})(3 \times 10^8)}{5460 \times 10^{-10}} \). The equation holds true, confirming  our understanding.

Step 3: Solving for Tungsten's Threshold Wavelength

Now, use the same formula for tungsten: Given \( \phi_{\text{W}} = 4.5 \mathrm{eV} \) and solve for \( \lambda_{\text{W}} \). Rearrange the formula: \( \lambda_{\text{W}} = \frac{hc}{\phi_{\text{W}}} \). Substitute: \( \lambda_{\text{W}} = \frac{(4.1357 \times 10^{-15})(3 \times 10^8)}{4.5} \). Calculate \( \lambda_{\text{W}} \).

Step 4: Calculation and Comparison

Calculate \( \lambda_{\text{W}} = \frac{(4.1357 \times 10^{-15})(3 \times 10^8)}{4.5} \approx 2769 \times 10^{-10} \mathrm{m} = 2769 \mathrm{\AA} \). Compare with the options provided: The closest value to our calculation is option (a) 2791 \mathrm{\AA}.

Key Concepts

Work Function

The work function is a fundamental concept in the photoelectric effect. It represents the minimum energy needed to remove an electron from the surface of a material. Think of it as the energy barrier that electrons must overcome to escape. This value is specific to each material, as different materials hold onto their electrons with varying strengths. The stronger the attraction between the nucleus and the electrons, the higher the work function.

• In our example, tungsten has a work function of 4.5 eV, while sodium has a work function of 2.3 eV.
• Therefore, electrons are more tightly bound to tungsten compared to sodium, requiring more energy to be emitted.

In practical terms, the work function helps in determining which materials are suitable for applications that rely on the photoelectric effect, such as photodetectors and solar cells. It also highlights how different wavelengths of light interact with different materials. That leads us to the next concept, the threshold wavelength.

Threshold Wavelength

The threshold wavelength is a critical parameter in understanding the photoelectric effect. It refers to the maximum wavelength, or lowest frequency, of light that can cause the ejection of an electron from a material's surface. It's essentially the point at which light has just enough energy to overcome the material's work function.

• Remember, as wavelength increases, the energy of the light decreases, because energy and wavelength are inversely related.
• Using the formula \[ \phi = \frac{hc}{\lambda} \]where \( \phi \) is the work function, we can calculate the threshold wavelength for different materials.

In the case of sodium, the threshold wavelength is given as \( 5460 \text{ Å} \). This means that light with a longer wavelength than this will not have enough energy to overcome the work function of sodium. For tungsten, since the work function is higher, the threshold wavelength is shorter, requiring more energetic light to emit electrons. This illustrates how the work function and threshold wavelength are intertwined.

Planck's Constant

Planck's constant plays a vital role in the realm of quantum mechanics and is at the heart of understanding the photoelectric effect. Its value is approximately \( 4.1357 \times 10^{-15} \text{ eV}\cdot \text{s} \), and it links the energy of photons (particles of light) to their frequency.In the context of the photoelectric effect:

• Planck's constant \( h \) is used in the equation \[ E = h \times u \]which correlates energy \( E \) with frequency \( u \).
• It also appears in the formula \[ \phi = \frac{hc}{\lambda} \]where it helps to connect the work function \( \phi \) with the threshold wavelength \( \lambda \).

In essence, Planck's constant allows us to measure and relate the discrete quantities of energy lost or gained during the interaction of light with matter. It fundamentally illustrates that energy transfers are not continuous but rather occur in specific quantities called quanta. This insight was pivotal in the development of both the photoelectric effect as a concept and the broader theory of quantum mechanics.