A photelectric material having work-function `phi_(0)` is illuminated with light of wavelength `lamda(lamdalt(hc)/(lamda_(0))).` The fastest photoelectron has a de Broglie wevelength `lamda_(d).` A change in wavelength of the incident light by `Deltalamda` results in a change `Deltalamda_(d)` in `lamda_(d).` Then the ratio `Deltalamda_(d)//Deltalamda` is proportional to

Options

A. `(lamda_d^3)/(lamda^2)`
B. `(lamda_d^3)/(lamda)`
C. `(lamda_d^2)/(lamda^2)`
D. `(lamda_d)/(lamda)`

Answer

To solve the problem, we will follow these steps:

### Step 1: Understand the Photoelectric Effect
The maximum kinetic energy (KE) of the emitted photoelectron can be expressed as:
\[ KE = \frac{1}{2} mv_{max}^2 = E_{incident} - \phi_0 \]
where \(E_{incident} = \frac{hc}{\lambda}\) is the energy of the incident photon and \(\phi_0\) is the work function of the material.

### Step 2: Express the Kinetic Energy in Terms of de Broglie Wavelength
The de Broglie wavelength \(\lambda_d\) of the photoelectron is given by:
\[ \lambda_d = \frac{h}{p} \]
where \(p\) is the momentum of the electron. The kinetic energy can also be expressed in terms of momentum:
\[ KE = \frac{p^2}{2m} \]
Substituting \(p = \frac{h}{\lambda_d}\) into the kinetic energy equation gives:
\[ KE = \frac{h^2}{2m\lambda_d^2} \]

### Step 3: Set Up the Equation for Maximum Kinetic Energy
Equating the two expressions for kinetic energy, we have:
\[ \frac{h^2}{2m\lambda_d^2} = \frac{hc}{\lambda} - \phi_0 \]

### Step 4: Differentiate the Equation
To find the relationship between changes in wavelength, we differentiate both sides with respect to \(\lambda\):
\[ \frac{d}{d\lambda}\left(\frac{h^2}{2m\lambda_d^2}\right) = \frac{d}{d\lambda}\left(\frac{hc}{\lambda} - \phi_0\right) \]

### Step 5: Apply the Chain Rule
Using the chain rule on the left side:
\[ \frac{d}{d\lambda}\left(\frac{h^2}{2m\lambda_d^2}\right) = -\frac{h^2}{m} \cdot \frac{1}{\lambda_d^3} \cdot \frac{d\lambda_d}{d\lambda} \]
On the right side:
\[ \frac{d}{d\lambda}\left(\frac{hc}{\lambda}\right) = -\frac{hc}{\lambda^2} \]

### Step 6: Set the Derivatives Equal
Setting the derivatives equal gives:
\[ -\frac{h^2}{m} \cdot \frac{1}{\lambda_d^3} \cdot \frac{d\lambda_d}{d\lambda} = -\frac{hc}{\lambda^2} \]

### Step 7: Solve for the Ratio of Changes
Rearranging the equation to find the ratio \(\frac{d\lambda_d}{d\lambda}\):
\[ \frac{d\lambda_d}{d\lambda} = \frac{m \cdot c \cdot \lambda^2}{h \cdot \lambda_d^3} \]

### Step 8: Find the Ratio \(\frac{\Delta \lambda_d}{\Delta \lambda}\)
The ratio \(\frac{\Delta \lambda_d}{\Delta \lambda}\) is proportional to:
\[ \frac{\Delta \lambda_d}{\Delta \lambda} \propto \frac{m \cdot c \cdot \lambda^2}{h \cdot \lambda_d^3} \]

### Conclusion
Thus, the ratio \(\frac{\Delta \lambda_d}{\Delta \lambda}\) is proportional to \(\frac{\lambda^2}{\lambda_d^3}\).