Two spherical objects with a mass of 7.60 kg each are placed at a distance of 2.93 m apart. How many electrons need to leave each object so that the net force between them becomes zero?
consider forces
kQ^2/distance^2=GM^2/distance^2
solve for q, then
number electrons=q/e
To calculate the number of electrons needed to leave each object in order to make the net force between them zero, we need to consider the electric force between the objects and the charge of an electron.
The electric force between two charged objects can be calculated using Coulomb's Law:
F = k * ((q1 * q2) / r^2)
Where:
F is the electric force between the objects,
k is the electrostatic constant (k = 9 x 10^9 N m^2/C^2),
q1 and q2 are the charges of the two objects, and
r is the distance between the two objects.
In this case, the net force between the objects should be zero. This means that the electric force between them should be balanced out by the repulsive force generated when electrons leave each object.
To balance out the electric force, the charge on each object should be such that the repulsive force generated by the charges of the electrons leaving is equal in magnitude and opposite in direction to the electric force.
Let's assume that each object initially has the same number of electrons. The total charge on each object will be the charge of one electron times the number of electrons:
q = n * e
Where:
q is the charge on the object,
n is the number of electrons, and
e is the charge of one electron (e = 1.6 x 10^-19 C).
Since the objects have the same mass and are at the same distance apart, we can assume that the electric force between them will be the same.
Therefore, we can set up an equation to balance the electric force and the force generated by the electrons leaving:
k * ((q1 * q2) / r^2) = F_repulsive
Since the objects have the same charge, we can substitute q1 with q/2 and q2 with -q/2:
k * (((q/2) * (-q/2)) / r^2) = F_repulsive
Simplifying this equation, we can find the value of q:
k * (q^2 / (4 * r^2)) = F_repulsive
q^2 = 4 * r^2 * (F_repulsive / k)
q = √(4 * r^2 * (F_repulsive / k))
Now we can substitute the given values:
F_repulsive = F (electric force) = k * ((q1 * q2) / r^2) = k * ((q/2) * (-q/2)) / r^2) = - (k * (q^2) / (4 * r^2))
F = 0 (net force is zero)
0 = - (k * (q^2) / (4 * r^2))
Simplifying this equation, we can solve for q:
q = √(-(4 * r^2 * F / k))
Now we can substitute the known values:
r = 2.93 m
F = 0 (net force is zero)
k = 9 x 10^9 N m^2/C^2
q = √(-(4 * (2.93 m)^2 * 0 / (9 x 10^9 N m^2/C^2)))
Calculating this equation, we find that q = 0.
Since q = n * e, we can conclude that no electrons need to leave each object in order to make the net force between them zero.