How far apart should two protons be if the electrical force of repulsion is equal to the weight (attraction to the Earth) of one of the protons?
m g = k q^2/r^2
1.67 * 10^-27 * 9.81 = 9*10^9 (1.6*10^-19)/r^2
To determine how far apart two protons should be if the electrical force of repulsion is equal to the weight (attraction to the Earth) of one of the protons, we need to equate the electrical force of repulsion to the gravitational force of attraction.
The electrical force of repulsion can be calculated using Coulomb's law:
Fe = (k * q1 * q2) / r^2
Where:
Fe is the electrical force of repulsion between the protons,
k is the electrostatic constant (approximately 9 × 10^9 N·m^2/C^2),
q1 and q2 are the charges of the protons (which are both positive since they repel each other),
r is the distance between the protons.
The weight of an object can be calculated using the formula:
W = m * g
Where:
W is the weight of the object,
m is the mass of the object,
g is the acceleration due to gravity (approximately 9.8 m/s^2 on Earth).
In this case, we know that the weight of one proton should be equal to the electrical force of repulsion. Since the masses of the protons are very small compared to their charges, we can assume that the masses of the protons are equal. Therefore, we can equate the weight to the electrical force:
m * g = (k * q1 * q2) / r^2
Now, solving for r, we have:
r^2 = (k * q1 * q2) / (m * g)
Taking the square root of both sides, we get:
r = sqrt((k * q1 * q2) / (m * g))
By substituting the known values of the relevant variables in the equation above, we can calculate the distance (r) between the two protons.
To calculate the distance between two protons at which the electrical force of repulsion is equal to the weight (attraction to Earth), we can equate the electrical force and gravitational force.
The electrical force between two protons can be calculated using Coulomb's Law, which states:
F_electric = (k * q^2) / r^2
Where:
F_electric is the electrical force of repulsion
k is the electrostatic constant (9 x 10^9 N*m^2/C^2)
q is the charge of each proton (which is equal to 1.6 x 10^-19 C)
r is the distance between the protons
The gravitational force between one proton and the Earth can be calculated using Newton's Law of Universal Gravitation, which states:
F_gravity = (G * m1 * m2) / r^2
Where:
F_gravity is the gravitational force
G is the gravitational constant (6.67 x 10^-11 N*m^2/kg^2)
m1 and m2 are the masses of the objects (in this case, the mass of one proton and the mass of Earth)
r is the distance between the proton and Earth's center (which can be assumed as the Earth's radius, approximately 6.37 x 10^6 m)
Since we want the electrical force to be equal to the weight, we can equate the two forces:
F_electric = F_gravity
(k * q^2) / r^2 = (G * m1 * m2) / r^2
Substituting the values for k, q, G, and the masses of the proton and Earth, we get:
(9 * 10^9 N*m^2/C^2 * (1.6 * 10^-19 C)^2) / r^2 = (6.67 * 10^-11 N*m^2/kg^2 * (1.67 * 10^-27 kg) * (5.97 * 10^24 kg)) / r^2
Simplifying the equation, we find:
r^2 = ((9 * 10^9 N*m^2/C^2 * (1.6 * 10^-19 C)^2) * r^2) / ((6.67 * 10^-11 N*m^2/kg^2 * (1.67 * 10^-27 kg) * (5.97 * 10^24 kg)))
r^2 = (2.304 * 10^-44 kg*m^7/s^2) / (5.58 * 10^-6 kg*m^2*s^-2)
r^2 = 4.128 * 10^-39 m^5*s^-2
Taking the square root of both sides, we get:
r ≈ 2.03 * 10^-8 m
Therefore, the distance between the two protons should be approximately 2.03 x 10^-8 meters to have the electrical force of repulsion equal to the weight (attraction to Earth) of one proton.