Let p and E denote the linear momentum and energy of a photon. If the wavelength is decreased,

Options

A. both p and E increase
B. p increases and E decreases
C. p decreases and E increases
D. both p and E decrease.

Answer

To solve the problem, we need to analyze the relationships between the wavelength of a photon, its energy, and its momentum.

### Step-by-Step Solution:

1. **Understanding the Relationships**:
- The energy \( E \) of a photon is given by the formula:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
- The momentum \( p \) of a photon is given by the formula:
\[ p = \frac{h}{\lambda} \]

2. **Analyzing the Effect of Decreasing Wavelength**:
- If the wavelength \( \lambda \) is decreased, we can observe the effects on both energy and momentum.
- For energy:
- Since \( E \) is inversely proportional to \( \lambda \), a decrease in \( \lambda \) will lead to an increase in \( E \).
- For momentum:
- Similarly, since \( p \) is also inversely proportional to \( \lambda \), a decrease in \( \lambda \) will lead to an increase in \( p \).

3. **Conclusion**:
- Therefore, when the wavelength is decreased, both the energy \( E \) and the momentum \( p \) of the photon increase.

4. **Final Answer**:
- The correct option is that both momentum and energy increase.