The number of photons of light having wave number 'x' in 10 J of energy source is :

Options

A. `10hcx`
B. `(hc)/(10x)`
C. `10/(hcx)`
D. none of these

Answer

To solve the problem of finding the number of photons of light having wave number 'x' in a 10 J energy source, we can follow these steps:

### Step-by-Step Solution:

1. **Understanding the Relationship Between Energy and Photons**:
The energy (E) of a photon can be expressed as:
\[ E = n \cdot h \cdot c \cdot \bar{\nu} \]
where:
- \(E\) = total energy (in joules)
- \(n\) = number of photons
- \(h\) = Planck's constant (\(6.626 \times 10^{-34} \, \text{J s}\))
- \(c\) = speed of light (\(3.00 \times 10^8 \, \text{m/s}\))
- \(\bar{\nu}\) = wave number (in m\(^{-1}\))

2. **Relating Wave Number to Wavelength**:
The wave number \(\bar{\nu}\) is defined as:
\[ \bar{\nu} = \frac{1}{\lambda} \]
where \(\lambda\) is the wavelength. Therefore, we can express energy in terms of wave number:
\[ E = n \cdot h \cdot c \cdot \bar{\nu} \]

3. **Substituting Known Values**:
We are given that the total energy \(E = 10 \, \text{J}\) and the wave number \(\bar{\nu} = x\). Thus, we can rewrite the equation:
\[ 10 = n \cdot h \cdot c \cdot x \]

4. **Rearranging to Solve for \(n\)**:
To find the number of photons \(n\), we rearrange the equation:
\[ n = \frac{10}{h \cdot c \cdot x} \]

5. **Final Expression**:
Therefore, the number of photons of light having wave number \(x\) in 10 J of energy is:
\[ n = \frac{10}{h \cdot c \cdot x} \]