In an X-ray tube, the relationship between voltage and wavelength is crucial for understanding how X-rays are produced. When you apply a voltage across the tube, it accelerates electrons towards a target material, and when these high-energy electrons collide with the target, they can produce X-rays. The cut-off wavelength is determined by the maximum energy of the electrons, which is directly related to the voltage applied. When you mention that only one of the wavelengths has characteristic emission, it implies that one of the voltages is sufficient to produce X-rays with specific energies associated with the target material, while the other voltage may not reach that threshold. Let's break this down further.

Understanding Cut-off Wavelengths

The cut-off wavelength (\( \lambda_c \)) can be calculated using the formula:

• \( \lambda_c = \frac{hc}{eV} \)

Here, \( h \) is Planck's constant, \( c \) is the speed of light, \( e \) is the elementary charge, and \( V \) is the voltage applied. This equation shows that as the voltage increases, the cut-off wavelength decreases, meaning higher energy X-rays are produced.

Characteristic Emission Explained

Characteristic X-rays are emitted when an electron from an inner shell of an atom is ejected, and an electron from a higher energy level falls into that lower energy state. This transition releases energy in the form of X-ray photons with specific wavelengths unique to the element of the target material. If only one of the applied voltages results in characteristic emission, it indicates that:

• The voltage is high enough to overcome the binding energy of the inner-shell electrons.
• The other voltage is either too low to produce these emissions or only generates bremsstrahlung radiation, which is continuous and not characteristic.

Implications of the Scenario

Given this context, if we consider the options (a, b, c, d) you mentioned, we can infer that:

• If one voltage produces characteristic X-rays, it must be greater than or equal to the threshold voltage required for that specific emission.
• The other voltage, which does not produce characteristic X-rays, must be lower than this threshold.

Therefore, the statement that is always true in this scenario is that the voltage producing characteristic emission is greater than the voltage that does not. This leads us to conclude that:

• The voltage associated with the characteristic emission is always higher than the voltage that does not produce such emissions.

In summary, understanding the relationship between voltage and the resulting X-ray emissions is key to interpreting the behavior of X-ray tubes. The ability to produce characteristic X-rays is contingent upon reaching a certain energy threshold, which is dictated by the applied voltage.