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Chapter 27: Problem 15

If the shortest wavelength produced by an x-ray tube is \(0.46 \mathrm{nm},\) what is the voltage applied to the tube?

Short Answer

Expert verified
Answer: The voltage applied to the x-ray tube is approximately 86.2 kV.

Step by step solution

01

Write down the Duane-Hunt law formula

Write down the Duane-Hunt law, which gives the relation between the minimum wavelength produced by an x-ray tube and the voltage applied to the tube: \(\lambda_{min} = \frac{hc}{eV}\)
02

Plug in the values for the constants

Insert the values for the Planck constant \(h \approx 6.626 \times 10^{-34} \mathrm{Js}\), the speed of light \(c \approx 3.0 \times 10^{8} \mathrm{m/s}\), and elementary charge \(e \approx 1.6 \times 10^{-19} \mathrm{C}\): \(0.46 \times 10^{-9} = \frac{(6.626 \times 10^{-34})(3.0 \times 10^{8})}{(1.6 \times 10^{-19})V}\)
03

Solve for V

Now, we will solve for voltage \(V\) applied across the tube. First, multiply both sides of the equation by \((1.6 \times 10^{-19})V\) to isolate the voltage term: \((0.46 \times 10^{-9})(1.6 \times 10^{-19})V = (6.626 \times 10^{-34})(3.0 \times 10^{8})\) Now, divide both sides of the equation by \((0.46 \times 10^{-9})(1.6 \times 10^{-19})\) to get \(V\) alone: \(V = \frac{(6.626 \times 10^{-34})(3.0 \times 10^{8})}{(0.46 \times 10^{-9})(1.6 \times 10^{-19})}\) Use a calculator to compute the voltage \(V\): \(V \approx 8.62 \times 10^{4} V\) So, the voltage applied to the x-ray tube is approximately \(8.62 \times 10^{4} V\) or \(86.2 \mathrm{kV}\).

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Most popular questions from this chapter

The minimum energy required to remove an electron from a metal is 2.60 eV. What is the longest wavelength photon that can eject an electron from this metal?
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