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 magnitude of the electrical force between two charged particles is given by the formula:
F = (k * |q1 * q2|) / r^2
where F is the electrical force, k is the electrostatic constant (approximately 9 x 10^9 N m^2/C^2), q1 and q2 are the charges of the particles, and r is the distance between them.
In this case, both particles are protons, which have equal charges of +1.602 x 10^-19 C (charge of an electron). The distance between them is 3.80 x 10^-10 m.
Substituting these values into the formula, we have:
F = (9 x 10^9 N m^2/C^2 * |1.602 x 10^-19 C * 1.602 x 10^-19 C|) / (3.80 x 10^-10 m)^2
Calculating this expression, we find that the electrical force exerted by one proton on the other is approximately 1.441 x 10^-8 N (newtons).
(b) To compare the magnitude of the electrical force with the magnitude of the gravitational force between the protons, we can use the formula for the gravitational force between two point masses:
F_gravity = (G * m1 * m2) / r^2
where F_gravity is the gravitational force, G is the gravitational constant (approximately 6.674 x 10^-11 N m^2/kg^2), m1 and m2 are the masses of the particles, and r is the distance between them.
Since the masses of protons are equal to approximately 1.673 x 10^-27 kg, we can substitute the values into the formula:
F_gravity = (6.674 x 10^-11 N m^2/kg^2 * 1.673 x 10^-27 kg * 1.673 x 10^-27 kg) / (3.80 x 10^-10 m)^2
Evaluating this equation, we find that the gravitational force between the protons is approximately 2.360 x 10^-68 N.
Therefore, to compare the magnitudes of these forces, we can divide the electrical force by the gravitational force:
(electrical force / gravitational force) = (1.441 x 10^-8 N) / (2.360 x 10^-68 N)
Simplifying this expression, we get a very large number which is approximately 6.10 x 10^59.
(c) If 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 these forces equal to each other:
(G * m1 * m2) / r^2 = (k * q1 * q2) / r^2
Since the distance r^2 cancels out, we can simplify the equation to:
G * m1 * m2 = k * q1 * q2
Rearranging the equation and solving for the charge-to-mass ratio (q/m), we have:
(q1 * q2) / (m1 * m2) = (G * k)^(-1/2)
Substituting the known values for G and k, we can calculate q/m:
(q1 * q2) / (m1 * m2) = ((6.674 x 10^-11 N m^2/kg^2) * (9 x 10^9 N m^2/C^2))^(-1/2)
Calculating this expression, we find that the charge-to-mass ratio in order for the magnitudes of the gravitational and electrical forces to be equal is approximately 8.16 x 10^14 C/kg.
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 given by:
F = k * (q1 * q2) / r^2
Where:
- F is the electrical force
- k is the electrostatic constant (k = 8.99 × 10^9 N m^2/C^2)
- q1 and q2 are the charges of the particles in coulombs
- r is the distance between the particles in meters
In this case, the charges of both protons are identical and equal to the elementary charge, which is approximately 1.6 × 10^-19 C. The distance between them is given as 3.80 × 10^-10 m.
(a) Magnitude of Electrical Force: Plugging in these values into Coulomb's law, we get:
F = (8.99 × 10^9 N m^2/C^2) * (1.6 × 10^-19 C)^2 / (3.80 × 10^-10 m)^2
Calculating this, we find:
F ≈ 2.30 × 10^-8 N
So, the electrical force exerted by one proton on the other is approximately 2.30 × 10^-8 N.
(b) Magnitude Comparison with Gravitational Force: The magnitude of the gravitational force between two particles can be calculated using Newton's law of universal gravitation:
F_gravity = G * (m1 * m2) / r^2
Where:
- F_gravity is the gravitational force
- G is the gravitational constant (G = 6.67430 × 10^-11 N m^2/kg^2)
- m1 and m2 are the masses of the particles in kilograms
Since protons have a very small mass (approximately 1.673 × 10^-27 kg), the gravitational force between them is negligible compared to the electrical force, so:
(electrical force / gravitational force) ≈ (2.30 × 10^-8 N) / (negligible) = ∞
The magnitude of the electrical force is infinitely greater than the magnitude of the gravitational force.
(c) Charge-to-Mass Ratio for Equal Magnitude Forces: To find the charge-to-mass ratio for which the magnitudes of the gravitational force and electrical force are equal, we set the two forces equal to each other:
G * (m1 * m2) / r^2 = k * (q1 * q2) / r^2
Cancelling out the (r^2) term, we have:
G * (m1 * m2) = k * (q1 * q2)
Rearranging the equation:
(q1 * q2) / (m1 * m2) = G / k
Substituting the known values:
(q1 * q2) / (1.6 × 10^-19 C * 1.6 × 10^-19 C) = (6.67430 × 10^-11 N m^2/kg^2) / (8.99 × 10^9 N m^2/C^2)
Simplifying it:
(q1 * q2) ≈ (1.6 × 10^-19 C * 1.6 × 10^-19 C) * (6.67430 × 10^-11 N m^2/kg^2) / (8.99 × 10^9 N m^2/C^2)
Calculating this, we find:
(q1 * q2) ≈ 1.1845 × 10^-29 C^2
Therefore, the particle's charge-to-mass ratio must be approximately 1.1845 × 10^-29 C^2/kg for the magnitudes of the gravitational and electrical forces to be equal.