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 increase and E decreases
C. p decrease and E increases
D. both p and E decrreases

Answer

To solve the problem, we need to analyze the relationship 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 can be expressed using the equation:
\[ E = \frac{hc}{\lambda} \]
where \( h \) is Planck's constant, \( c \) is the speed of light, and \( \lambda \) is the wavelength.
- The linear momentum (p) of a photon is given by:
\[ p = \frac{E}{c} \]
Substituting the expression for energy, we get:
\[ p = \frac{hc}{\lambda c} = \frac{h}{\lambda} \]

2. **Analyzing the Effect of Decreasing Wavelength**:
- If the wavelength \( \lambda \) is decreased, we can analyze how it affects energy and momentum:
- From the energy equation \( E = \frac{hc}{\lambda} \):
- As \( \lambda \) decreases, \( E \) increases (since \( E \) is inversely proportional to \( \lambda \)).
- From the momentum equation \( p = \frac{h}{\lambda} \):
- As \( \lambda \) decreases, \( p \) also increases (since \( p \) is inversely proportional to \( \lambda \)).

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

### Final Answer:
- The correct option is **A: both P and E increase**.