Two iron spheres each have a volume of 10 cm^3 and a mass of 78g. How many electrons must be placed on each sphere so that the electrostatic force of repulsion between them just balances the force of gravitational attraction? Assume the distance between the spheres is more than 2 meters.
You can treat the spheres as point charges at R >2m separation. If e is the charge on each sphere, for the electrical repulsion to equal the gravitational attraction,
G M^2 /R^2 = k Q^2/R^2
GM^2 = k Q^2
G is the Newton gravity constant and k is the Coulomb law constant
Solve for Q. The number of electrons is Q/e.
e is the elctron charge.
To find the number of electrons needed on each sphere, we need to calculate the electric charge on each sphere first.
Step 1: Calculate the mass of each sphere in kg.
Given mass of each sphere = 78g
Converting grams to kilograms: 1g = 0.001kg
Mass of each sphere = 78g * 0.001kg/g = 0.078kg
Step 2: Calculate the volume of each sphere in m^3.
Given volume of each sphere = 10cm^3
Converting cm^3 to m^3: 1cm^3 = (0.01m)^3 = 0.000001m^3
Volume of each sphere = 10cm^3 * 0.000001m^3/cm^3 = 0.00001m^3
Step 3: Calculate the density of each sphere.
Density = Mass/Volume
Density of each sphere = 0.078kg/0.00001m^3 = 7800kg/m^3
Step 4: Calculate the gravitational force of attraction between the spheres.
The force of gravitational attraction between two objects can be calculated using Newton's law of universal gravitation:
F_grav = (G * m1 * m2) / r^2
where F_grav is the force of gravity, G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between them.
Given that the distance between the spheres is more than 2 meters, we can assume a safe distance of 2 meters.
Gravitational force = (6.67430 × 10^-11 N m^2 / kg^2) * ((0.078kg)^2 / (2m)^2)
Step 5: Calculate the number of electrons needed on each sphere.
The electrostatic force of repulsion between two charged objects can be calculated using Coulomb's law:
F_elec = (k * q1 * q2) / r^2
where F_elec is the electrostatic force, k is the electrostatic constant, q1 and q2 are the charges on the two objects, and r is the distance between them.
We need to find the number of electrons (q) that will produce an electrostatic force equal to the gravitational force calculated in Step 4.
Setting F_grav = F_elec, we can solve for q1 or q2 (since q1 = q2).
(q1)^2 = (F_grav * r^2) / (k * m^2)
q1 = sqrt((F_grav * r^2) / (k * m^2))
where q1 is the charge on one sphere.
Electric charge on each sphere = q1 = sqrt((F_grav * r^2) / (k * m^2))
where r is the distance between the spheres, k is the electrostatic constant, F_grav is the gravitational force between the spheres, and m is the mass of each sphere.
Step 6: Calculate the number of electrons.
Electric charge of one electron = -1.6 × 10^-19 C
Number of electrons = charge on each sphere / charge of one electron
Number of electrons = electric charge of each sphere / (-1.6 × 10^-19 C)
By following these steps and plugging in the given values, you should be able to find the number of electrons needed on each sphere.