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
A satellite is orbiting Earth in an approximately circular path. It completes one revolution each day (86,400 seconds). Its orbital distance is 22,500 kilometers from the Earth's center. What is the satellite's average speed?
To solve this problem, we can use the principles of circular motion and the relationship between electric force and centripetal acceleration.
(a) The electric force exerted on an electron by a proton can be calculated using Coulomb's Law, which states that the magnitude of the electric force between two charged particles is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
The equation for the electric force is given by:
F = (k * q1 * q2) / r^2
Where F is the electric force, k is Coulomb's constant (9 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the particles (which are 1.6 x 10^-19 C for both the electron and the proton), and r is the distance between them (0.542 x 10^-10 m).
Plugging in the values, we can calculate the electric force exerted on each particle:
F = (9 x 10^9 Nm^2/C^2) * (1.6 x 10^-19 C)^2 / (0.542 x 10^-10 m)^2
Simplifying the equation will give us the value in Newtons (N).
(b) Since the electric force is causing the centripetal acceleration of the electron, we can equate the two forces:
F = m * a
Where F is the electric force, m is the mass of the electron (9.11 x 10^-31 kg), and a is the centripetal acceleration.
We know that the centripetal acceleration can be calculated using the formula:
a = v^2 / r
Where v is the speed of the electron and r is the radius of the orbit (0.542 x 10^-10 m).
Substituting this equation into the force equation, we get:
(9.11 x 10^-31 kg) * (v^2 / r) = F
Now, rearranging the equation to solve for v:
v^2 = (F * r) / (9.11 x 10^-31 kg)
Finally, taking the square root of both sides gives us the speed of the electron in meters per second (m/s):
v = √[(F * r) / (9.11 x 10^-31 kg)]
Calculating this equation with the appropriate values for F and r will give you the speed of the electron.