In the Bohr theory of the hydrogen atom, an electron moves in a circular orbit about a proton, where the radius of the orbit is approximately 0.542 multiplied by 10-10 m. (The actual value is 0.529 multiplied by 10-10 m.)
(a) Find the electric force exerted on each particle, based on the approximate (not actual) radius given.
N
(b) If this force causes the centripetal acceleration of the electron, what is the speed of the electron?
m/s
41
To answer the given questions, we can use the relationship between the electric force and the centripetal force acting on the electron.
(a) The electric force between the proton and electron can be found using Coulomb's law, which states that the magnitude of the electric force between two charged particles is given by:
F = (k * |q1 * q2|) / r^2
where F is the electric force, k is the electrostatic constant (9 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the particles (in this case, q1 = -e and q2 = e, where e is the elementary charge), and r is the radius of the orbit.
Substituting the given values, we have:
F = (9 x 10^9 Nm^2/C^2) * (e * e) / (0.542 x 10^-10 m)^2
Calculating this expression will give us the electric force exerted on each particle.
(b) The centripetal force experienced by the electron is provided by the electric force. The centripetal force can be calculated using the equation:
F = m * a_c
where F is the centripetal force, m is the mass of the electron, and a_c is the centripetal acceleration of the electron.
Since the force mentioned in part (a) is acting as the centripetal force, we can equate the two forces:
F = m * a_c
Solving this equation for a_c gives:
a_c = F / m
The centripetal acceleration can be written as:
a_c = (v^2) / r
where v is the speed of the electron.
Equating the two expressions for a_c, we have:
(v^2) / r = F / m
Solving for v:
v = √((F * r) / m)
Substituting the known values, including the value of the force obtained in part (a), and the mass of the electron, which is approximately 9.11 x 10^-31 kg, we can calculate the speed of the electron.