Electrons used in an electron microscope are accelerated by a voltage of 25 kV . If the voltage is increased to 100 kV then de-Broglie wavelength associated with the electrons would

Options

A. Increase by 4 times
B. Increase by 2 times
C. Decrease by 2 times
D. Decrease by 4 times

Answer

To find the change in the de Broglie wavelength associated with electrons when the accelerating voltage is increased from 25 kV to 100 kV, we can follow these steps:

### Step 1: Understand the relationship between de Broglie wavelength and voltage
The de Broglie wavelength (\( \lambda \)) of an electron is given by the formula:
\[ \lambda = \frac{h}{p} \]
where \( p \) is the momentum of the electron. The momentum can be expressed in terms of the kinetic energy gained by the electron when it is accelerated through a voltage \( V \):
\[ KE = eV \]
where \( e \) is the charge of the electron.

### Step 2: Relate kinetic energy to momentum
The kinetic energy can also be expressed in terms of momentum:
\[ KE = \frac{p^2}{2m} \]
where \( m \) is the mass of the electron. Setting these two expressions for kinetic energy equal gives:
\[ eV = \frac{p^2}{2m} \]
From this, we can express momentum \( p \) as:
\[ p = \sqrt{2m eV} \]

### Step 3: Substitute momentum into the de Broglie wavelength formula
Substituting the expression for momentum into the de Broglie wavelength formula, we get:
\[ \lambda = \frac{h}{\sqrt{2m eV}} \]

### Step 4: Analyze the effect of changing voltage
From the formula, we can see that the de Broglie wavelength is inversely proportional to the square root of the voltage:
\[ \lambda \propto \frac{1}{\sqrt{V}} \]

### Step 5: Calculate the change in wavelength
Let’s denote the initial voltage as \( V_1 = 25 \, \text{kV} \) and the final voltage as \( V_2 = 100 \, \text{kV} \).

The ratio of the wavelengths at these two voltages can be expressed as:
\[ \frac{\lambda_2}{\lambda_1} = \frac{\sqrt{V_1}}{\sqrt{V_2}} \]
Substituting the values:
\[ \frac{\lambda_2}{\lambda_1} = \frac{\sqrt{25}}{\sqrt{100}} = \frac{5}{10} = \frac{1}{2} \]

### Step 6: Conclusion
This means that when the voltage is increased from 25 kV to 100 kV, the de Broglie wavelength associated with the electrons is halved.

### Final Answer
The de Broglie wavelength associated with the electrons would decrease to half its original value.

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