(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, 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 for calculating the electrical force (F) is given by:
F = k * (q1 * q2) / r^2
Where:
F is the electrical force,
k is the electrostatic constant (9 x 10^9 N.m^2/C^2),
q1 and q2 are the charges of the particles (in this case, both charges are equal to the elementary charge e = 1.6 x 10^-19 C),
and r is the distance between the charges (3.80 x 10^-10 m).
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
Solving this equation 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 gravitational force between two protons, we need to calculate the gravitational force. The gravitational force between two objects can be calculated using Newton's Law of Universal Gravitation.
The formula for calculating the gravitational force (Fg) is given by:
Fg = G * (m1 * m2) / r^2
Where:
Fg is the gravitational force,
G is the gravitational constant (approximately 6.67 x 10^-11 N.m^2/kg^2),
m1 and m2 are the masses of the objects (in this case, the masses are equal to the proton mass, which is approximately 1.67 x 10^-27 kg),
and r is the distance between the masses (3.80 x 10^-10 m).
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
Solving this equation will give us the magnitude of the gravitational force exerted by one proton on the other.
To compare the magnitudes of the electrical force and the gravitational force, we can divide the electrical force by the gravitational force:
(F / Fg)
(c) To find the charge-to-mass ratio of a particle, where the magnitude of the gravitational force between two particles is equal to the magnitude of the electrical force between them, we can set the equations for electrical force (F) and gravitational force (Fg) equal to each other and solve for the charge-to-mass ratio (q/m).
(9 x 10^9 N.m^2/C^2) * ((q1 * q2) / r^2) = (6.67 x 10^-11 N.m^2/kg^2) * ((m1 * m2) / r^2)
Cancelling out common terms and rearranging the equation:
(q1 * q2) / (m1 * m2) = (6.67 x 10^-11 N.m^2/kg^2) / (9 x 10^9 N.m^2/C^2)
Simplifying further, we can substitute q = q1 = q2 and m = m1 = m2 for the charge and mass of the particle:
q^2 / m^2 = (6.67 x 10^-11 N.m^2/kg^2) / (9 x 10^9 N.m^2/C^2)
Taking the square root of both sides, we get:
q/m = √((6.67 x 10^-11 N.m^2/kg^2) / (9 x 10^9 N.m^2/C^2))
Simplifying this equation will give us the charge-to-mass ratio of the particle.