(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
(a) 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 equation is given by:
F = k * (q1 * q2) / (r^2)
Where:
F is the electrical force between the particles
k is the electrostatic constant, approximately 9 x 10^9 Nm^2/C^2
q1 and q2 are the charges of the particles (in this case both are protons, which have a charge of +1.6 x 10^-19 C)
r is the distance between the particles (3.80 x 10^-10 m)
Plugging in the values into the equation:
F = (9 x 10^9 Nm^2/C^2) * ((1.6 x 10^-19 C)^2) / ((3.80 x 10^-10 m)^2)
Evaluating the expression will give us the magnitude of the electrical force exerted by one proton on the other.
(b) To compare the magnitude of the electrical force with the magnitude of the gravitational force, we can calculate the gravitational force between the protons using Newton's law of universal gravitation. The equation is given by:
F_grav = G * (m1 * m2) / r^2
Where:
F_grav is the gravitational force between the protons
G is the gravitational constant, approximately 6.674 x 10^-11 Nm^2/kg^2
m1 and m2 are the masses of the particles (in this case both are protons, which have a mass of approximately 1.67 x 10^-27 kg)
r is the distance between the particles (3.80 x 10^-10 m)
Plugging in the values into the equation:
F_grav = (6.674 x 10^-11 Nm^2/kg^2) * ((1.67 x 10^-19 kg)^2) / ((3.80 x 10^-10 m)^2)
Evaluating the expression will give us the magnitude of the gravitational force exerted by one proton on the other.
To compare the magnitude of the two forces, we can divide the electrical force by the gravitational force:
(electrical force / gravitational force) = F / F_grav
(c) To find the charge-to-mass ratio that would make the magnitude of the gravitational force equal to the magnitude of the electrical force, we set the two forces equal to each other and solve for the charge-to-mass ratio.
Using the equations from parts (a) and (b), we have:
(k * (q1 * q2) / (r^2)) = (G * (m1 * m2) / r^2)
We can cancel out common terms and rearrange the equation to solve for the charge-to-mass ratio:
(q1 * q2) / (m1 * m2) = G / k
Plugging in the values for G and k, we can evaluate the right-hand side of the equation:
G / k = (6.674 x 10^-11 Nm^2/kg^2) / (9 x 10^9 Nm^2/C^2)
Simplifying the expression will give us the required charge-to-mass ratio in C/kg.