Question

An accelerator boosts a proton's kinetic energy so that its de Broglie wavelength is \(3.63 \cdot 10^{-15} \mathrm{~m}\). What is the total energy of the proton?

Short answer

Answer: The total energy of the proton is approximately \(1.01\cdot 10^{-10} \mathrm{~J}\).

Step-by-step solution

Use the de Broglie wavelength formula

The formula for the de Broglie wavelength is given by: \[ \lambda = \frac{h}{p} \] where \(\lambda\) is the de Broglie wavelength, \(h\) is the Planck's constant (approximately \(6.63\cdot10^{-34}\mathrm{~Js}\)), and \(p\) is the momentum of the particle. We have \(\lambda = 3.63 \cdot 10^{-15} \mathrm{~m}\). We can rearrange the formula and solve for the momentum, \(p\): \[ p = \frac{h}{\lambda} \]

Calculate the momentum of the proton

Use the values of \(\lambda\) and \(h\) to find the momentum of the proton: \[ p = \frac{6.63\cdot 10^{-34}\mathrm{~Js}}{3.63 \cdot 10^{-15} \mathrm{~m}} \] After calculating, we get: \[ p \approx 1.83 \cdot 10^{-19} \mathrm{ ~kg \cdot m/s} \]

Find the kinetic energy of the proton

To find the kinetic energy, we can use the relation between momentum and kinetic energy: \[ KE = \frac{p^2}{2m} \] where \(KE\) is the kinetic energy, \(p\) is the momentum, and \(m\) is the mass of the proton. The mass of a proton is approximately \(1.67\cdot 10^{-27} \mathrm{~kg}\). Plug in the values: \[ KE = \frac{(1.83\cdot 10^{-19} \mathrm{~kg \cdot m/s})^2}{2(1.67\cdot 10^{-27} \mathrm{~kg})} \] Then, we get: \[ KE \approx 5.10 \cdot 10^{-11} \mathrm{~J} \]

Calculate the total energy of the proton

The total energy of a particle is the sum of its kinetic energy and rest mass energy. The rest mass energy can be found using Einstein's famous equation: \[ E_0 = mc^2 \] where \(E_0\) is the rest mass energy, \(m\) is the mass of the proton, and \(c\) is the speed of light (\((3.00\cdot 10^8 \mathrm{~m/s})\)). Plug in the values: \[ E_0 = (1.67\cdot 10^{-27} \mathrm{~kg}) (3.00\cdot 10^8 \mathrm{~m/s})^2 \] We get: \[ E_0 \approx 1.50 \cdot 10^{-10} \mathrm{~J} \] Now, we can find the total energy by adding the kinetic energy and the rest mass energy: \[ E = E_0+ KE \approx (1.50\cdot 10^{-10} + 5.10\cdot 10^{-11})\mathrm{~J} \] Finally, the total energy of the proton is: \[ E \approx 1.01\cdot 10^{-10} \mathrm{~J} \]