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 given by the equation:
F = k * (q1 * q2) / r^2
where F is the electrical force, k is the electrostatic constant (9.0 x 10^9 Nm^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.
In this case, we have two protons, both of which have the same charge, q = 1.6 x 10^-19 C. The distance between them is given as 3.80 x 10^-10 m. Substituting these values into the formula, we get:
F = (9.0 x 10^9 Nm^2/C^2) * ((1.6 x 10^-19 C)*(1.6 x 10^-19 C)) / (3.80 x 10^-10 m)^2
Simplifying the calculation, we find:
F ≈ 2.3 x 10^-8 N
So, the electrical force exerted by one proton on the other is approximately 2.3 x 10^-8 N.
Moving on to the second part of the question, we need to compare the magnitudes of the electrical force and the gravitational force between the two protons. The gravitational force between two objects is given by the equation:
Fg = G * (m1 * m2) / r^2
where Fg is the gravitational force, G is the gravitational constant (6.67 x 10^-11 Nm^2/kg^2), m1 and m2 are the masses of the objects, and r is the distance between them.
Since we're dealing with protons, which have a mass of approximately 1.67 x 10^-27 kg, we can use this value for m1 and m2. Substituting the values into the gravitational force equation, we get:
Fg = (6.67 x 10^-11 Nm^2/kg^2) * ((1.67 x 10^-27 kg) * (1.67 x 10^-27 kg)) / (3.80 x 10^-10 m)^2
Simplifying the calculation, we find:
Fg ≈ 2.4 x 10^-50 N
Therefore, the magnitude of the electrical force (2.3 x 10^-8 N) is many orders of magnitude greater than the magnitude of the gravitational force (2.4 x 10^-50 N) between the two protons.
For the last part of the question, let's assume that we want the magnitude of the gravitational force to be equal to the magnitude of the electrical force between the two protons.
Setting F = Fg, we can solve for the charge-to-mass ratio (q/m) that will satisfy this condition. Rearranging the equations, we get:
(k * (q1 * q2) / r^2) = (G * (m1 * m2) / r^2)
Since q1 = q2 = q (charge of a proton) and m1 = m2 = m (mass of a proton), we can substitute these values into the equation:
(k * (q^2) / r^2) = (G * (m^2) / r^2)
Simplifying the equation, we find:
q/m = sqrt((G * (m^2)) / (k * (q^2)))
Using the known values for G, k, m, and q, we can substitute them into the formula and calculate the desired charge-to-mass ratio.