The resolving power of an electron microscope operated at 16kV is R. The resolving power of the electron microscope when operated at 4kV is

Options

A. `(R )/(4)`
B. `(R )/(2)`
C. `4R`
D. `2R`

Explanation

To solve the problem of finding the resolving power of an electron microscope when operated at 4 kV, given that the resolving power at 16 kV is R, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding Resolving Power**:
The resolving power (R) of an electron microscope is inversely proportional to the wavelength (λ) of the electrons used. Thus, we can express this relationship as:
\[ R \propto \frac{1}{\lambda} \]

2. **Relating Wavelength to Momentum**:
The wavelength of the electrons is given by the de Broglie wavelength formula:
\[ \lambda = \frac{h}{p} \]
where \( h \) is Planck's constant and \( p \) is the momentum of the electrons.

3. **Momentum and Voltage**:
The momentum \( p \) can be expressed in terms of the kinetic energy of the electrons. The kinetic energy (KE) of an electron accelerated through a potential difference \( E \) (in volts) is given by:
\[ KE = eE \]
where \( e \) is the charge of the electron. The momentum can be related to the kinetic energy as follows:
\[ p = \sqrt{2m \cdot KE} = \sqrt{2m \cdot eE} \]
where \( m \) is the mass of the electron.

4. **Resolving Power in Terms of Voltage**:
Since \( R \) is directly proportional to the momentum, we can express it as:
\[ R \propto \sqrt{E} \]
Therefore, we can write:
\[ R = k \sqrt{E} \]
where \( k \) is a proportionality constant.

5. **Setting Up the Equations**:
For the two different voltages, we can write:
- For \( E_1 = 16 \, \text{kV} = 16000 \, \text{V} \):
\[ R_1 = k \sqrt{16000} \]
- For \( E_2 = 4 \, \text{kV} = 4000 \, \text{V} \):
\[ R_2 = k \sqrt{4000} \]

6. **Finding the Ratio of Resolving Powers**:
To find the relationship between \( R_1 \) and \( R_2 \), we can set up the ratio:
\[ \frac{R_1}{R_2} = \frac{k \sqrt{16000}}{k \sqrt{4000}} = \frac{\sqrt{16000}}{\sqrt{4000}} \]
Simplifying this gives:
\[ R_2 = R_1 \cdot \frac{\sqrt{16000}}{\sqrt{4000}} = R \cdot \frac{\sqrt{16}}{\sqrt{4}} = R \cdot \frac{4}{2} = R \cdot 2 \]

7. **Final Result**:
Therefore, the resolving power of the electron microscope when operated at 4 kV is:
\[ R_2 = \frac{R}{2} \]

### Conclusion:
The resolving power of the electron microscope when operated at 4 kV is \( \frac{R}{2} \).