Question

What is the shortest wavelength \(\mathrm{x}\) -ray produced by a \(0.20-\mathrm{MV}\) x-ray machine?

Short answer

Answer: The shortest wavelength of the x-ray produced by the 0.20-MV x-ray machine is approximately 62 pm.

Step-by-step solution

Identify the given information and the formula to be used

We are given the following information: - Voltage (V) = 0.20 MV = 0.20 x 10^6 V We need to find the wavelength (λ) of the shortest x-ray produced by the machine. The relationship between a photon's energy (E) and its wavelength (λ) is given by the formula: E = h * c / λ, where h is Planck's constant (6.63 x 10^(-34) Js), c is the speed of light (3.0 x 10^8 m/s), and λ is the wavelength. When an electron is accelerated through a potential difference (V), it gains an energy E = e * V, where e is the elementary charge (1.6 x 10^(-19) C).

Set up an equation for the x-ray's wavelength

To find the x-ray's shortest wavelength, we have to equate the energy gained by the electron to the energy of the x-ray: e * V = h * c / λ

Solve for the x-ray's wavelength

Now, rearrange the equation from step 2 and solve for λ: λ = (h * c) / (e * V) Substitute the given values and constants into the equation: λ = (6.63 x 10^(-34) Js * 3.0 x 10^8 m/s) / (1.6 x 10^(-19) C * 0.20 x 10^6 V) λ ≈ 6.2 x 10^(-11) m

Convert the wavelength to the desired unit

The shortest wavelength of the x-ray produced by the 0.20-MV x-ray machine is 6.2 x 10^(-11) m. This can also be expressed as: - 62 pm (picometers), since 1 m = 10^(12) pm So, the shortest wavelength of the x-ray produced by this machine is approximately 62 pm.