Electron are emited from an electron gun at almost zero velocity and are accelerated by an electric field E through a distance of `1.0m` The electron are now scatteared by an atomic hydrogen sample in ground state what should be the minimum value of E so that red light of wavelength `656.5 nm` may be emitted by the hydrogen?

Explanation

To solve the problem, we need to determine the minimum electric field \( E \) required to emit red light of wavelength \( 656.5 \, \text{nm} \) from a hydrogen atom after the electrons have been accelerated through a distance of \( 1.0 \, \text{m} \).

### Step-by-Step Solution:

1. **Identify the Wavelength and Energy Relation**:
The wavelength of the emitted light is given as \( \lambda = 656.5 \, \text{nm} \). We can convert this to meters:
\[ \lambda = 656.5 \times 10^{-9} \, \text{m} \]

2. **Calculate the Energy of the Photon**:
The energy \( E \) of a photon can be calculated using the formula:
\[ E = \frac{hc}{\lambda} \]
where \( h = 6.626 \times 10^{-34} \, \text{Js} \) (Planck's constant) and \( c = 3 \times 10^8 \, \text{m/s} \) (speed of light). Plugging in the values:
\[ E = \frac{(6.626 \times 10^{-34})(3 \times 10^8)}{656.5 \times 10^{-9}} \]
\[ E \approx 3.02 \times 10^{-19} \, \text{J} \]

3. **Convert Energy to Electron Volts**:
To convert energy from joules to electron volts (1 eV = \( 1.6 \times 10^{-19} \, \text{J} \)):
\[ E \approx \frac{3.02 \times 10^{-19}}{1.6 \times 10^{-19}} \approx 1.89 \, \text{eV} \]

4. **Determine the Minimum Energy Required**:
The minimum energy required to excite the electron from the ground state (n=1) to n=3 (since the transition to n=2 does not emit red light) can be calculated using the formula:
\[ E = 13.6 \left( \frac{1}{n_1^2} - \frac{1}{n_2^2} \right) \]
Here, \( n_1 = 1 \) and \( n_2 = 3 \):
\[ E = 13.6 \left( \frac{1}{1^2} - \frac{1}{3^2} \right) = 13.6 \left( 1 - \frac{1}{9} \right) = 13.6 \left( \frac{8}{9} \right) \approx 12.09 \, \text{eV} \]

5. **Calculate the Voltage Required**:
The potential difference \( V \) needed to provide this energy to the electron is given by:
\[ V = E \]
Therefore, \( V \approx 12.09 \, \text{V} \).

6. **Calculate the Electric Field \( E \)**:
The electric field \( E \) can be calculated using the formula:
\[ E = \frac{V}{d} \]
where \( d = 1.0 \, \text{m} \):
\[ E = \frac{12.09}{1} = 12.09 \, \text{V/m} \]

### Final Answer:
The minimum value of the electric field \( E \) required is:
\[ E \approx 12.09 \, \text{V/m} \]