What is the de Broglie wavelength of an electron that has been accelerated through a potential difference of 1.50 V?

This de Broglie wavelength calculator will help you describe the wave-particle duality of matter. According to this principle, we can analyze light as waves, explaining its properties such as refraction. On the other hand, light must be made of particles (called photons) if we want to understand the photoelectric effect. De Broglie stated that every particle could be described as having a particular wavelength and formulated the famous de Broglie equation. If you want to know the de Broglie wavelength of an electron or a photon, keep reading!

According to de Broglie, a beam of particles of some mass can behave as a matter wave. Its wavelength is related to the mass and velocity of the particle: λ = h / (m × v),

where:

• m is the mass of the particle;
• v is the velocity of the particle, and
• h is the Planck constant, equal to 6.6261×10-34 Js.

The mass of particles is usually given in kilograms, but using scientific notation since the numbers are small. In the calculator, use the drop-down menu to select the exponent of the mass. For reference, these are the value of the electron rest mass (me) and the atomic mass unit (u), which is an average of the proton and neutron masses.

1 me = 9.10938356 × 10-31 kg

1 u = 1.660538921 × 10-27 kg

Let's find the de Broglie wavelength of an electron traveling at 1% of the speed of light.

1. The mass of an electron is 1 me, or
   9.10938356*10-31 kg
2. The speed of this electron is
   299,792,458 m/s / 100 = 2,997,924.58 m/s.
3. Multiplying the mass and speed, we obtain the momentum:
   p = mv = 2.7309245 × 10-24 kg·m/s
4. Dividing the Planck constant by p, we obtain the de Broglie wavelength:
   h/p = 6.6261*10-34 / 2.7309245*10-24 = 2.426 × 10-10 m.
5. Hence, the de Broglie wavelength of our electron is 0.24 nm.

Since a photon's rest mass is zero, you might be wondering how to use our de Broglie wavelength calculator to find the de Broglie wavelength of a photon. Thanks to wave-particle duality, a photon does have a momentum associated with it. The value of this momentum can be used to calculate the de Broglie wavelength of the photon.

A question asks you to find the de Broglie wavelength of a photon which has a momentum of 6.8 × 10-35 kg·m/s. Let's see how to use our calculator to answer this question.

1. Click on the unit changer menu of the momentum row and select x10-35 from the list.
2. Enter 6.8 into the momentum row. You've now entered the correct value for the momentum of the photon.
3. You will see the result for the wavelength in nanometers by default. As it is a huge value, you'll probably want to change the units to something more manageable, such as meters. The result is 9.74 m, rounding to 3 significant figures.

The unit in which the de Broglie wavelength is expressed is meters. Since the de Broglie wavelength is usually very, very small, we most often express this value in nanometres.

To determine the de Broglie wavelength of a particle given its mass and velocity, you need to:

1. Multiply the velocity by mass. Their product is the particle's momentum.
2. Divide Planck's constant by the momentum found in Step 1.
3. The result you've got is exactly the de Broglie wavelength of your particle. Congrats!

QUESTION: Approximately what is the "de Broglie" wavelength of an electron that has been accelerated through a potential difference of = 150 V? (The mass of an electron = 9.11 x 10-31 kg.) Possible answers: (a) 0.1 nm (b) 1 nm (c) 10 nm (d) 100 nm (e) 1000 nm The kinetic energy of an electron accelerated through a potential difference of V volts is given by the equation:

� mv2 = eV where e is the electron charge (1.6x10-19 C) [You must be given the electron charge and Planck's constant in order to answer this question].

and so v = √(2eV/m) = 7.26x106 m/s

Therefore: wavelength = h/mv = h/[9.11x10- 31x7.26x106] = 1x10-10 = 0.1 nm

The correct answer is (a)