one electron travels twice as fast as another. Which has the longer wavelength?
The De Broglie wavelength of matter is inversely proportional to momentum and velocity.
The slower electron has the longer wavelength.
To determine which electron has a longer wavelength, we can utilize the equation
wavelength = Planck's constant / (mass * velocity),
where Planck's constant is a fundamental constant in physics (approximately 6.62607015 × 10^-34 m² kg / s), mass refers to the mass of a single electron (approximately 9.10938356 × 10^-31 kilograms), and velocity represents the velocity of each electron. Let's label the velocity of the first electron as v1 and the velocity of the second electron as v2.
Given that the first electron travels twice as fast as the second electron (v1 = 2v2), we can calculate the wavelengths of each electron and compare them.
For the first electron:
wavelength1 = Planck's constant / (mass * v1)
For the second electron:
wavelength2 = Planck's constant / (mass * v2)
Substituting v1 = 2v2 into the equation for the first electron's wavelength, we have:
wavelength1 = Planck's constant / (mass * 2v2)
= 1/2 * (Planck's constant / (mass * v2))
Comparing the two wavelengths, we observe that wavelength1 = 1/2 * wavelength2. This means that the first electron has half the wavelength of the second electron.
Therefore, the electron traveling twice as fast has the shorter wavelength since wavelength and velocity are inversely proportional.
To determine which electron has the longer wavelength, we can use the de Broglie wavelength equation, which relates the wavelength of a particle to its velocity. The equation is:
λ = h / mv,
Where:
λ is the wavelength of the wave associated with the particle,
h is the Planck's constant (approximately 6.626 x 10^-34 Js),
m is the mass of the particle, and
v is the velocity of the particle.
Since the two electrons have different velocities, we can compare their wavelengths by calculating the ratios of their velocities. Let's assume the velocity of one electron is v1, and the velocity of the other electron is v2, where v1 = 2v2.
To find the ratio of their wavelengths, we can divide the equation for v1 by the equation for v2:
v1 / v2 = 2v2 / v2 = 2.
Therefore, the ratio of their velocities is 2.
Now, let's substitute the relation between the velocities into the de Broglie equation to find the ratio of their wavelengths:
λ1 / λ2 = (h / mv1) / (h / mv2)
= (h / m) * (v2 / v1)
= (h / m) * (1 / 2).
We can see that the ratio of their wavelengths is 1/2. Since the ratio is less than 1, the electron with the higher velocity (v2) has the shorter wavelength. Thus, the electron that travels twice as fast has the shorter wavelength.