In a sample of hydrogen-like atom all of which are in ground state, a photon beam containing photos of various energies is passed. In absorption spectrum, five dark lines, are observed. The number of bright lines in the emission spectrum will be (assume that all transitions takes place).

Options

A. `5`
B. `10`
C. `15`
D. None of these

Explanation

To solve the problem, we need to determine the number of bright lines in the emission spectrum based on the information provided about the absorption spectrum of a hydrogen-like atom.

### Step-by-Step Solution:

1. **Understanding the Absorption Spectrum**:
- The problem states that there are 5 dark lines observed in the absorption spectrum.
- The number of dark lines in the absorption spectrum is given by the formula:
\[ \text{Number of dark lines} = n - 1 \]
where \( n \) is the number of energy levels that the electrons can transition to from the ground state.

2. **Finding the Number of Energy Levels (n)**:
- Given that there are 5 dark lines, we can set up the equation:
\[ n - 1 = 5 \]
- Solving for \( n \):
\[ n = 5 + 1 = 6 \]

3. **Understanding the Emission Spectrum**:
- The number of bright lines in the emission spectrum is given by the formula:
\[ \text{Number of bright lines} = \frac{n(n - 1)}{2} \]
- Now we can substitute \( n = 6 \) into this formula:
\[ \text{Number of bright lines} = \frac{6(6 - 1)}{2} \]

4. **Calculating the Number of Bright Lines**:
- Continuing with the calculation:
\[ \text{Number of bright lines} = \frac{6 \times 5}{2} = \frac{30}{2} = 15 \]

5. **Conclusion**:
- Therefore, the number of bright lines in the emission spectrum is **15**.

### Final Answer:
The number of bright lines in the emission spectrum will be **15**.

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