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
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
To find the electric force exerted on each particle in the Bohr model, we can use Coulomb's Law, which states that the electric force between two charged particles is given by:
F = (k * |q1 * q2|) / r^2
where F is the force, k is the electrostatic constant (k = 9 x 10^9 N m^2/C^2), q1 and q2 are the charges of the two particles, and r is the distance between them.
In this case, we have an electron (q1) moving in a circular orbit around a proton (q2), and the radius of the orbit is given as 0.542 x 10^-10 m.
(a) Finding the electric force:
The charges of the electron and proton are equal in magnitude but opposite in sign, so we can simplify the equation to:
F = (k * |e|^2) / r^2
where e is the elementary charge (e = 1.6 x 10^-19 C).
Substituting the given values into the equation, we have:
F = (9 x 10^9 N m^2/C^2 * (1.6 x 10^-19 C)^2) / (0.542 x 10^-10 m)^2
Calculating this expression will give us the electric force exerted on each particle.
(b) Finding the speed of the electron:
The electric force provides the centripetal force required to keep the electron in a circular orbit. The centripetal force is given by:
F = mv^2 / r
where m is the mass of the electron and v is its speed.
Since the electric force calculated in part (a) is equal to the centripetal force, we can equate the two equations:
(k * |e|^2) / r^2 = mv^2 / r
Simplifying this equation, we have:
v = √((k * |e|^2) / mr)
Now, we can substitute the known values (k, e, m, and r) into this equation to find the speed of the electron.