Question

When accelerated electrons are directed against an anticathode in an X-ray tube, the radiation obtained has a continuous spectrum with a wavelength minimum, \(\lambda_{\min }=\frac{1.24 \times 10^{-6}}{V} \mathrm{~m}\), where \(\mathrm{V}\) is the volt- age used for accelerating the electrons. \(\lambda_{\min }\) for \(V=5 \times 10^{4} \mathrm{~V}\) is (a) \(0.124 \mathrm{~nm}\) (b) \(0.248 \mathrm{~nm}\) (c) \(2.48 \mathrm{~nm}\) (d) \(1.24 \mathrm{~nm}\)

Short answer

\(0.248 \mathrm{~nm}\)

Step-by-step solution

Understanding the Equation

Acknowledge the given equation for the minimum wavelength, \(\frac{1.24 \times 10^{-6}}{V} \mathrm{~m}\), which describes the relationship between the minimum wavelength \(\lambda_{\min}\) of the X-ray radiation and the voltage \(V\) applied to accelerate the electrons.

Substituting the Given Values

Substitute the given value of the voltage \(V = 5 \times 10^{4} \mathrm{~V}\) into the equation to find the minimum wavelength \(\lambda_{\min}\).

Calculating the Minimum Wavelength

Perform the calculation to determine the minimum wavelength: \(\lambda_{\min} = \frac{1.24 \times 10^{-6}}{5 \times 10^{4}} \mathrm{~m}\).

Convert the Wavelength to Nanometers

Convert the resulting wavelength in meters to nanometers (1 m = 10^9 nm) for comparison with the answer choices.

Verifying the Correct Answer

Verify which of the given options corresponds to the calculated minimum wavelength in nanometers.

Key concepts

Continuous X-ray Spectrum

X-rays, much like visible light, can possess a spectrum that varies continuously in wavelength. However, unlike the discrete emission spectrum of atoms in a gas, which occurs at specific wavelengths, the X-ray spectrum stretches across a broad range without distinct lines. This is mainly due to the way X-rays are produced. When electrons are accelerated and strike a metal target (the anticathode), the sudden deceleration of these electrons due to the atomic nuclei produces a broad range of X-ray wavelengths. This range is termed a continuous spectrum because the emitted X-rays can have any wavelength down to a certain minimum, with intensities that taper off as the wavelength decreases. This minimum is directly related to the energy of the accelerated electrons, which is determined by the voltage applied to move these electrons.

It's essential to realize that the continuous X-ray spectrum represents numerous possible X-ray photons with varying energies, and no two X-ray tubes will produce the exact same spectrum. Factors that can modify the shape of the continuous spectrum include the voltage applied to the tube and the material of the target. For students, picturing the X-ray spectrum as a smooth curve descending from a peak value down to a point at the minimum wavelength where the intensity falls to zero can be helpful in understanding this concept.

Accelerated Electrons

Accelerated electrons are central to the generation of X-rays in an X-ray tube. These high-speed electrons are produced by heating a filament to release electrons, a process known as thermionic emission, similar to the way old-fashioned light bulbs work. Once the electrons are free from the filament, they are accelerated towards a positive metal target by a very high voltage. During this process, their increase in kinetic energy is directly proportional to the voltage applied; the higher the voltage, the faster they travel and the more kinetic energy they gain.

Upon impact with the metal target, the electrons are rapidly decelerated, and their kinetic energy is converted into X-rays. A significant insight here is that not all the kinetic energy converts into a single photon of X-ray. Rather, it contributes to the continuous spectrum, with a variety of photon energies being emitted. For a comprehensive understanding, consider that a fraction of the electron's energy can also be transformed into heat within the target, which is a factor when designing X-ray systems to prevent overheating.

Wavelength-Voltage Relationship

The minimum wavelength of X-ray radiation produced in an X-ray tube is inversely related to the voltage used to accelerate the electrons, as described by the equation given in the exercise. This relationship is vital for understanding how changing the voltage affects the X-rays emitted. As the voltage increases, the electrons gain more kinetic energy before striking the target, which translates to the production of higher energy X-rays, hence shorter minimum wavelength. This can be seen in the equation ewlineewline\[\begin{equation} \lambda_{\min} = \frac{1.24 \times 10^{-6}}{V} \mathrm{~m} \end{equation}\]where the constant (1.24 x 10^-6) incorporates both physical constants and converts the units to meters. By using this formula, one can determine the shortest wavelength of X-ray that can be produced for any given voltage. This information is particularly useful in medical imaging and materials analysis where varying the wavelength of X-rays can provide different levels of penetration and resolution. This equation, therefore, provides a direct method to tailor the X-rays' properties to specific applications merely by adjusting the voltage in the X-ray tube.