The mass of an electron is 9.11\times 10^{-31}~\rm kg. If the de Broglie wavelength for an electron in a hydrogen atom is 3.31\times 10^{-10}~\rm m, how fast is the electron moving relative to the speed of light? The speed of light is 3.00\times10^8 ~\rm m/s.
w = wavelength
w = h/mv with m in kg.
Solve for velocity in m/s.
Then v/3E8 = velocity relative to c.
To solve this question, we can use the de Broglie wavelength equation, which relates the wavelength of a particle to its momentum:
λ = h / p
where λ is the de Broglie wavelength, h is the Planck's constant (6.63 x 10^-34 J·s), and p is the momentum of the particle.
We can rearrange this equation to solve for the momentum (p):
p = h / λ
Now, we know that the momentum of an object is given by:
p = mv
where m is the mass of the object and v is its velocity. Rearranging the equation again, we get:
v = p / m
Substituting the given values into the equations, we get:
p = (6.63 x 10^-34 J·s) / (3.31 x 10^-10 m) = 2 x 10^-24 N·s
v = (2 x 10^-24 N·s) / (9.11 x 10^-31 kg) = 2.19 x 10^6 m/s
Finally, to find the speed of the electron relative to the speed of light, we divide the calculated velocity by the speed of light:
v_relative = (2.19 x 10^6 m/s) / (3 x 10^8 m/s) = 7.29 x 10^-3
Therefore, the electron is moving at approximately 0.729% of the speed of light.