What additional energy must be supplied to the electric

What additional energy must be imparted to an electron with a momentum of 15 keV/s (c is the speed of light) so that its de Broglie wavelength becomes equal to 50 nm?

Problem 50325. Detailed solution with a brief record of the conditions, formulas and laws used in the solution, derivation of the calculation formula and answer. If you have any questions regarding the solution, please write. I try to help.

To solve this problem it is necessary to use the de Broglie formula:

λ = h / p

where λ is the de Broglie wavelength, h is Planck’s constant, p is the electron momentum.

From the problem conditions it is known that the electron momentum is 15 keV/s. It is necessary to find what additional energy needs to be imparted to the electron so that its de Broglie wavelength becomes equal to 50 nm.

First, you need to express the momentum of the electron in terms of energy:

p = √(2mE)

where m is the electron mass, E is the energy.

Thus we have:

15 keV/s = √(2 * 9.11 * 10^-31 * E)

From here we find the energy value:

E = (15 keV/s)^2 / (2 * 9.11 * 10^-31) = 3.061 * 10^-15 J

Next, using de Broglie’s formula, we find the initial de Broglie wavelength:

λ₀ = h / p₀ = 6.626 * 10^-34 J s / (√(2 * 9.11 * 10^-31 * 3.061 * 10^-15 J)) = 0.122 nm

The found de Broglie wavelength is 50 nm times less than the desired one, therefore, it is necessary to impart additional energy to the electron.

To find the additional energy, it is necessary to substitute the desired value of the de Broglie wavelength λ = 50 nm and the found value of the momentum p₀ into the de Broglie formula:

50 nm = 6.626 * 10^-34 J s / p + 6.626 * 10^-34 J s / (√(2 * 9.11 * 10^-31 * 3.061 * 10^-15 J))

Solving the equation for p we get:

p = 1.67 * 10^-24 kg m/s

Now you can find the value of the additional energy:

ΔE = (p^2 - p₀^2) / (2m) = 4.71 * 10^-15 J

So, in order for the de Broglie wavelength of an electron to become equal to 50 nm, it is necessary to provide it with an additional energy of 4.71 * 10^-15 J.

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From the existing description it is not entirely clear what exactly needs to be described. If we are talking about a product that needs to be described, then it is necessary to clarify what kind of product we are talking about. If we are talking about what additional energy needs to be imparted to the electron so that it can perform some functions, then the answer depends on the specific situation.

For example, if we are talking about charging a battery, then it is necessary to provide enough energy to charge the battery to the desired level. When it comes to the functioning of an electronic device, it is necessary to provide a sufficient level of voltage and current so that the device can operate in the desired mode.

In general, for each specific case it is necessary to carry out calculations and determine what additional energy needs to be imparted to the electron.

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