For harder X-rays,

Options

A. the wavelength is higher
B. the intensity is higher
C. the frequency is heigher
D. the photon energy is higher

Answer

To solve the question regarding harder X-rays, we need to analyze the relationship between the wavelength, frequency, and photon energy of X-rays when the accelerating voltage is increased.

### Step-by-Step Solution:

1. **Understanding Harder X-rays**:
- Harder X-rays refer to X-rays with higher energy and shorter wavelengths.

2. **Effect of Accelerating Voltage (V)**:
- The energy of the X-ray photons is directly related to the accelerating voltage. When the voltage (V) is increased, the energy of the X-rays produced also increases.

3. **Cutoff Wavelength (λ_min)**:
- The minimum wavelength (cutoff wavelength) of the X-rays can be calculated using the formula:
\[ \lambda_{\text{min}} = \frac{hc}{eV} \]
- Here, \(h\) is Planck's constant, \(c\) is the speed of light, \(e\) is the charge of an electron, and \(V\) is the accelerating voltage.
- From this formula, we can see that as \(V\) increases, \(\lambda_{\text{min}}\) decreases.

4. **Relationship Between Wavelength and Frequency**:
- The frequency (\(ν\)) of the X-rays is related to the wavelength by the equation:
\[ c = λν \quad \Rightarrow \quad ν = \frac{c}{λ} \]
- Since \(\lambda\) decreases as \(V\) increases, the frequency \(ν\) must increase.

5. **Photon Energy**:
- The energy (\(E\)) of a photon is given by:
\[ E = hν \]
- Since we established that frequency \(ν\) increases with increasing voltage, the energy \(E\) of the photons also increases.

6. **Conclusion**:
- Therefore, for harder X-rays, both the frequency and the photon energy are higher. The correct options from the given choices are:
- Option 3: The frequency is higher.
- Option 4: The photon energy is higher.

### Final Answer:
- The correct options for harder X-rays are: **Option 3 (the frequency is higher)** and **Option 4 (the photon energy is higher)**.