Two photons having

Options

A. equal wavelengths have equal linear momenta
B. equal energies have equal linear momenta
C. equal frequencies have equal linear momenta
D. equal linear momenta have equal wavelengths.

Answer

To solve the question regarding the relationship between photons and their properties, we will analyze each option step by step.

### Step 1: Understand the Properties of Photons
Photons are particles of light that have both energy and momentum. The energy (E) of a photon is given by the equation:
\[ E = h\nu \]
where \( h \) is Planck's constant and \( \nu \) is the frequency of the photon. The momentum (P) of a photon can be expressed as:
\[ P = \frac{E}{c} \]
where \( c \) is the speed of light.

### Step 2: Analyze Option 1
**Option 1:** Two photons having equal wavelengths have equal linear momenta.

- If two photons have equal wavelengths (\( \lambda_1 = \lambda_2 \)), then their frequencies are also equal because:
\[ \nu = \frac{c}{\lambda} \]
- Since the frequencies are equal, their energies will also be equal:
\[ E_1 = E_2 \]
- Consequently, their momenta will be equal:
\[ P_1 = P_2 \]
- However, momentum is a vector quantity, and while the magnitudes may be equal, the directions can differ. Thus, this option is **incorrect**.

### Step 3: Analyze Option 2
**Option 2:** Two photons having equal energies have equal linear momenta.

- If the energies are equal (\( E_1 = E_2 \)), then using the momentum formula:
\[ P_1 = \frac{E_1}{c} \quad \text{and} \quad P_2 = \frac{E_2}{c} \]
- Since \( E_1 = E_2 \), it follows that \( P_1 = P_2 \) in magnitude. However, as before, the direction of the momentum may differ. Therefore, this option is also **incorrect**.

### Step 4: Analyze Option 3
**Option 3:** Two photons having equal frequencies have equal linear momenta.

- If the frequencies are equal (\( \nu_1 = \nu_2 \)), then their energies are equal:
\[ E_1 = h\nu_1 \quad \text{and} \quad E_2 = h\nu_2 \]
- Thus, their momenta will also be equal:
\[ P_1 = \frac{E_1}{c} \quad \text{and} \quad P_2 = \frac{E_2}{c} \]
- Again, while the magnitudes are equal, the directions may differ. Hence, this option is **incorrect**.

### Step 5: Analyze Option 4
**Option 4:** Two photons having equal linear momenta have equal wavelengths.

- If the linear momenta are equal (\( P_1 = P_2 \)), then their energies must also be equal:
\[ P = \frac{E}{c} \Rightarrow E_1 = E_2 \]
- Since the energies are equal, their wavelengths must also be equal because:
\[ E = h\nu = \frac{hc}{\lambda} \]
- Therefore, if \( E_1 = E_2 \), it follows that \( \lambda_1 = \lambda_2 \). This option is **correct**.

### Conclusion
The correct answer is **Option 4**: Two photons having equal linear momenta have equal wavelengths.

---