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
F=k•q₁•q₂/r²
k =9•10⁹ N•m²/C²
e =1.6•10⁻¹⁹ C.
F=k•e²/r²=
=9•10⁹•(1.6•10⁻¹⁹)²/(0.542•10⁻¹⁰)² =
=7.84•10⁻⁸ N
ma=mv²/R= F
v=sqrt(FR/m)=
=sqrt(7.84•10⁻⁸•0.542•10⁻¹⁰/9.1•10⁻³¹)=
=2.16•10⁶ m/s
To find the electric force exerted on each particle, we can use Coulomb's law. Coulomb's law states that the electric force between two charged particles is given by:
F = k * (q1 * q2) / r^2
Where:
- F is the electric force between the particles
- k is the electrostatic constant (k ≈ 9 × 10^9 N m^2/C^2)
- q1 and q2 are the magnitudes of the charges on the particles
- r is the distance between the particles
In this case, the two particles are the electron and the proton. The charge of an electron is -1.6 × 10^-19 C, and the charge of a proton is +1.6 × 10^-19 C (the magnitudes of the charges are the same). The given radius is approximately 0.542 × 10^-10 m.
(a) To find the electric force exerted on each particle, we can substitute the values into Coulomb's law:
F = (9 × 10^9 N m^2/C^2) * ((1.6 × 10^-19 C) * (1.6 × 10^-19 C)) / (0.542 × 10^-10 m)^2
Simplifying the calculation, we have:
F = (9 × 10^9 N m^2/C^2) * (2.56 × 10^-38 C^2) / (2.95 × 10^-20 m^2)
F = (2.304 × 10^-28 N m^2/C^2) / (2.95 × 10^-20 m^2)
F = 7.818 × 10^-9 N
Therefore, the approximate electric force exerted on each particle is 7.818 × 10^-9 N.
(b) The electric force between the electron and the proton provides the centripetal force required to maintain the circular orbit of the electron. The centripetal force is given by:
F = m * a
Where:
- F is the centripetal force
- m is the mass of the electron
- a is the centripetal acceleration
Since F is the same as the electric force we calculated above, we can equate the two:
7.818 × 10^-9 N = m * a
The electron's mass is approximately 9.11 × 10^-31 kg.
To find the speed of the electron, we need to find the centripetal acceleration. The centripetal acceleration is given by:
a = v^2 / r
Where:
- a is the centripetal acceleration
- v is the velocity (speed) of the electron
- r is the radius of the orbit
Substituting this into the equation above, we have:
7.818 × 10^-9 N = (9.11 × 10^-31 kg) * (v^2 / (0.542 × 10^-10 m))
Simplifying the calculation, we have:
7.818 × 10^-9 N * (0.542 × 10^-10 m) = 9.11 × 10^-31 kg * v^2
v^2 ≈ (7.818 × 10^-9 N * (0.542 × 10^-10 m)) / (9.11 × 10^-31 kg)
v^2 ≈ 4.663 × 10^11 m^2/s^2
Taking the square root of both sides:
v ≈ √(4.663 × 10^11 m^2/s^2)
v ≈ 6.825 × 10^5 m/s
Therefore, the approximate speed of the electron is 6.825 × 10^5 m/s.