(a) Two protons in a molecule are 3.80 multiplied by 10-10 m apart. Find the electrical force exerted by one proton on the other.
Magnitude
N
(b) State how the magnitude of this force compares with the magnitude of the gravitational force exerted by one proton on the other.
(electrical force / gravitational force)
(c) What if? What must be a particle's charge-to-mass ratio if the magnitude of the gravitational force between two of these particles is equal to the magnitude of electrical force between them?
C/kg
To solve these questions, we'll use Coulomb's Law, which states that the electrical force between two charged particles is proportional to the product of their charges and inversely proportional to the square of the distance between them. The formula is as follows:
F = (k * |q1 * q2|) / r^2
Where:
F is the magnitude of the electrical force.
k is the Coulomb's constant, which is approximately 9 x 10^9 N * m^2 / C^2.
q1 and q2 are the charges of the particles.
r is the distance between the particles.
Now let's solve each question:
(a) We are given that two protons are 3.80 x 10^-10 m apart. The charge of a proton is 1.6 x 10^-19 C. Since both protons have the same charge, the charges in the formula are q1 = q2 = 1.6 x 10^-19 C. Plugging in the values:
F = (9 x 10^9 N * m^2 / C^2) * |(1.6 x 10^-19 C) * (1.6 x 10^-19 C)| / (3.80 x 10^-10 m)^2
Calculating this gives us:
F ≈ 2.30 x 10^-8 N
So, the magnitude of the electrical force exerted by one proton on the other is approximately 2.30 x 10^-8 N.
(b) To compare the magnitudes of electrical and gravitational forces, we need to find the gravitational force exerted by one proton on the other. The formula for gravitational force is given by:
Fg = (G * |m1 * m2|) / r^2
Where:
Fg is the magnitude of the gravitational force.
G is the gravitational constant, which is approximately 6.67 x 10^-11 N * m^2 / kg^2.
m1 and m2 are the masses of the particles.
r is the distance between the particles.
Since mass and charge are unrelated, the mass of a proton is approximately 1.67 x 10^-27 kg. Plugging in the values:
Fg = (6.67 x 10^-11 N * m^2 / kg^2) * (1.67 x 10^-27 kg) * (1.67 x 10^-27 kg) / (3.80 x 10^-10 m)^2
Calculating this gives us:
Fg ≈ 3.76 x 10^-48 N
So, the magnitude of the gravitational force exerted by one proton on the other is approximately 3.76 x 10^-48 N.
To compare the magnitudes of the forces:
electrical force / gravitational force = (2.30 x 10^-8 N) / (3.76 x 10^-48 N)
Calculating this gives us:
electrical force / gravitational force ≈ 6.13 x 10^39
So, the magnitude of the electrical force is significantly larger than the magnitude of the gravitational force.
(c) To find the charge-to-mass ratio for the gravitational force to be equal to the electrical force, we can set the two forces equal to each other and solve for the charge-to-mass ratio:
(k * |q1 * q2|) / r^2 = (G * |m1 * m2|) / r^2
Since the distance (r) is the same for both forces, we can cancel it out:
k * |q1 * q2| = G * |m1 * m2|
Now, substitute the known values:
(9 x 10^9 N * m^2 / C^2) * (q1 * q2) = (6.67 x 10^-11 N * m^2 / kg^2) * (m1 * m2)
Simplifying:
(q1 * q2) = (6.67 x 10^-11 N * m^2 / kg^2) * (m1 * m2) / (9 x 10^9 N * m^2 / C^2)
Substitute the known values:
(1.6 x 10^-19 C) * (1.6 x 10^-19 C) = (6.67 x 10^-11 N * m^2 / kg^2) * (1.67 x 10^-27 kg) / (9 x 10^9 N * m^2 / C^2)
Calculating this gives us:
(1.6 x 10^-19 C)^2 ≈ 2.89 x 10^-47 kg / C
So, the particle's charge-to-mass ratio must be approximately 2.89 x 10^-47 C/kg for the magnitude of the gravitational force between two particles to be equal to the magnitude of the electrical force between them.