In one joule of energy, the number of photons with wave number equal to x is

Options

A. `(hcx)^(-1)`
B. `x(hc)^(-1)`
C. hcx
D. `hc(x)^(-1)`

Explanation

To solve the question of how many photons correspond to one joule of energy with a given wave number \( x \), we can follow these steps:

### Step-by-Step Solution:

1. **Understand the Definitions**:
- The energy of a photon is given by the equation:
\[ E = n \cdot h \cdot \nu \]
where \( E \) is the total energy, \( n \) is the number of photons, \( h \) is Planck's constant, and \( \nu \) is the frequency of the photon.

2. **Relate Frequency to Wave Number**:
- The wave number \( x \) is defined as:
\[ x = \frac{1}{\lambda} \]
where \( \lambda \) is the wavelength. The frequency \( \nu \) can be expressed in terms of the speed of light \( c \) and the wavelength \( \lambda \):
\[ \nu = \frac{c}{\lambda} \]

3. **Substitute Frequency in the Energy Equation**:
- Substitute \( \nu \) into the energy equation:
\[ E = n \cdot h \cdot \frac{c}{\lambda} \]

4. **Express Wavelength in Terms of Wave Number**:
- Since \( x = \frac{1}{\lambda} \), we can express \( \lambda \) as:
\[ \lambda = \frac{1}{x} \]

5. **Substitute Wavelength into the Energy Equation**:
- Substitute \( \lambda \) back into the energy equation:
\[ E = n \cdot h \cdot c \cdot x \]

6. **Rearrange to Solve for Number of Photons \( n \)**:
- Rearranging gives:
\[ n = \frac{E}{h \cdot c \cdot x} \]

7. **Substitute the Given Energy**:
- Given that \( E = 1 \) joule, we can substitute this into the equation:
\[ n = \frac{1}{h \cdot c \cdot x} \]

### Final Answer:
The number of photons with wave number equal to \( x \) in one joule of energy is given by:
\[ n = \frac{1}{h \cdot c \cdot x} \]