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, we can use Coulomb's law, which states that the electric force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them.
(a) The formula for 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 (k ≈ 9 x 10^9 N m^2 / C^2),
q1 and q2 are the charges of the particles, and
r is the distance between the particles.
In this case, we have a proton (q1) and an electron (q2). The electron is negatively charged, so q2 = -e, where e is the elementary charge (e ≈ 1.6 x 10^-19 C). The proton is positively charged, so q1 = +e.
The distance between the particles is given as r = 0.542 x 10^-10 m.
Plugging in the values into Coulomb's law, we have:
F = (k * |+e * -e|) / (0.542 x 10^-10)^2
Simplifying and calculating the expression, we get:
F ≈ 8.99 x 10^9 N m^2 / C^2 * 1.6 x 10^-19 C^2 / (0.542 x 10^-10 m)^2
F ≈ 8.99 x 1.6 / (0.542)^2 x 10^-10 / 10^-19 N
F ≈ 4.156 x 10^-8 N
So, the approximate electric force exerted on each particle is 4.156 x 10^-8 N.
(b) We can use the concept of centripetal force to find the speed of the electron. The centripetal force required to keep the electron in a circular orbit is provided by the electric force.
The centripetal force is given by:
F_c = (m * v^2) / r
Where:
F_c is the centripetal force,
m is the mass of the electron (m ≈ 9.11 x 10^-31 kg),
v is the speed of the electron, and
r is the radius of the orbit.
Since F_c is equal to the electric force (F), we can equate the two expressions:
4.156 x 10^-8 N = (9.11 x 10^-31 kg * v^2) / (0.542 x 10^-10 m)
Simplifying and rearranging the equation to solve for v, we get:
v^2 = (4.156 x 10^-8 N * 0.542 x 10^-10 m) / (9.11 x 10^-31 kg)
v^2 ≈ 2.458 x 10^3 m^2/s^2
Taking the square root gives us the speed of the electron:
v ≈ √(2.458 x 10^3 m^2/s^2)
v ≈ 49.57 x 10^3 m/s
So, the approximate speed of the electron is 49.57 x 10^3 m/s.