Two positive charged spheres, one with 2 times the charge of the other are 38 cm apart. The net force between them is 0.77 N. What is the charge on the lesser charged sphere?
To solve this problem, we can use Coulomb's Law, which relates the force between two charged objects to their charges and the distance between them.
Coulomb's Law states that the force between two charged objects is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.
Mathematically, it can be expressed as:
F = k * (q1 * q2) / r^2
where:
F is the force between the two objects,
k is the electrostatic constant (k = 9.0 x 10^9 N·m^2/C^2),
q1 and q2 are the charges on the two objects, and
r is the distance between the two objects.
In this problem, we have two positive charged spheres. Let's denote the charge on the lesser charged sphere as q1 and the charge on the other sphere as q2.
We are given that the distance between the spheres (r) is 38 cm, which is equivalent to 0.38 m.
The net force between the spheres (F) is given as 0.77 N.
Using Coulomb's Law, we can set up the equation:
0.77 N = (9.0 x 10^9 N·m^2/C^2) * (q1 * q2) / (0.38 m)^2
Now, let's solve for q1, the charge on the lesser charged sphere.
First, rearrange the equation:
q1 * q2 = (0.77 N * (0.38 m)^2) / (9.0 x 10^9 N·m^2/C^2)
Simplify the equation:
q1 * q2 = (0.77 N * 0.1444 m^2) / 9.0 x 10^9 N·m^2/C^2
q1 * q2 = 0.110888 / 9.0 x 10^9 C^2
Now, we need to determine q1. Since we know that the charge on the other sphere (q2) is 2 times the charge on q1, we can substitute q1 with 2q1:
(2q1) * q2 = 0.110888 / 9.0 x 10^9 C^2
2q1 * q2 = 0.110888 / 9.0 x 10^9 C^2
Since q2 = 2q1, we can substitute q2 in terms of q1:
2q1 * (2q1) = 0.110888 / 9.0 x 10^9 C^2
4q1^2 = 0.110888 / 9.0 x 10^9 C^2
Now, solve for q1:
q1^2 = (0.110888 / 9.0 x 10^9 C^2) / 4
q1^2 = 0.110888 / (4 * 9.0 x 10^9 C^2)
q1^2 = 0.0030804444 / 3.6 x 10^10 C^2
q1^2 ≈ 8.5568 x 10^-14 C^2
Taking the square root of both sides:
q1 ≈ 9.253 x 10^-7 C
Therefore, the charge on the lesser charged sphere (q1) is approximately 9.253 x 10^-7 Coulombs.