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 find the electrical force exerted by one proton on the other, we can use Coulomb's Law. Coulomb's Law states that the electrical 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 formula for the electrical force is:
F = k * (q1 * q2) / r^2
where F is the electrical force, k is the electrostatic constant, q1 and q2 are the charges of the two particles, and r is the distance between them.
The electrostatic constant, k, has a value of 9 × 10^9 Nm²/C².
In this case, we have two protons, each with a charge of +1.6 × 10^-19 C.
The distance between the protons is 3.80 × 10^-10 m.
Plugging in these values into the formula:
F = (9 × 10^9 Nm²/C²) * ((1.6 × 10^-19 C) * (1.6 × 10^-19 C)) / (3.80 × 10^-10 m)^2
Calculating this:
F ≈ 2.3 × 10^-27 N
So, the magnitude of the electrical force exerted by one proton on the other is approximately 2.3 × 10^-27 N.
Now, let's compare this magnitude with the magnitude of the gravitational force exerted by one proton on the other.
The formula for the gravitational force is:
F = G * (m1 * m2) / r^2
where F is the gravitational force, G is the gravitational constant, m1 and m2 are the masses of the two particles, and r is the distance between them.
The gravitational constant, G, has a value of approximately 6.67 × 10^-11 Nm²/kg².
Each proton has a mass of approximately 1.67 × 10^-27 kg.
Plugging in these values into the formula:
F = (6.67 × 10^-11 Nm²/kg²) * ((1.67 × 10^-27 kg) * (1.67 × 10^-27 kg)) / (3.80 × 10^-10 m)^2
Calculating this:
F ≈ 1.4 × 10^-67 N
So, the magnitude of the gravitational force exerted by one proton on the other is approximately 1.4 × 10^-67 N.
To compare the magnitudes of the electrical and gravitational forces, we can divide the electrical force by the gravitational force:
(electrical force / gravitational force) = (2.3 × 10^-27 N) / (1.4 × 10^-67 N)
Simplifying this:
(electrical force / gravitational force) ≈ 1.6 × 10^40
Therefore, the magnitude of the electrical force between the protons is approximately 1.6 × 10^40 times greater than the magnitude of the gravitational force between them.
Finally, if the magnitude of the gravitational force between two particles is equal to the magnitude of the electrical force between them, the charge-to-mass ratio of the particles must be equal to the gravitational constant divided by the electrostatic constant:
(charge-to-mass ratio) = G / k
Plugging in the values of G and k:
(charge-to-mass ratio) = (6.67 × 10^-11 Nm²/kg²) / (9 × 10^9 Nm²/C²)
Calculating this:
(charge-to-mass ratio) ≈ 7.4 × 10^-20 C/kg
So, for the magnitudes of the gravitational and electrical forces to be equal, the charge-to-mass ratio of the particles must be approximately 7.4 × 10^-20 C/kg.