I've been working on this problem for quite some time now and still can not figure it out...
Initially, both metal spheres are neutral.
In a charging process, 1 × 1013 electrons are
removed from one metal sphere and placed on
a second sphere. Then the electrical poten-
tial energy associated with the two spheres is
found to be −0.061 J .
The Coulomb constant is 8.98755 ×
109 N · m2/C2 and the charge on an electron
is 1.6 × 10−19 C.
What is the distance between the two
spheres?
Answer in units of m.
To solve this problem, we can use the formula for electric potential energy:
Electric potential energy (U) = (k * q1 * q2) / r
Where:
- k is the Coulomb constant (k = 8.98755 x 10^9 N · m^2/C^2)
- q1 and q2 are the charges of the two spheres
- r is the distance between the spheres
In this case, q1 is the charge removed from the first sphere (-1 x 10^13 electrons) and q2 is the charge added to the second sphere (+1 x 10^13 electrons).
First, we need to calculate the charges in Coulombs:
Charge in Coulombs = (charge in electrons) * (charge of an electron)
q1 = (-1 x 10^13) * (1.6 x 10^-19 C) = -1.6 x 10^-6 C
q2 = (1 x 10^13) * (1.6 x 10^-19 C) = 1.6 x 10^-6 C
Substituting the values into the formula:
-0.061 J = (8.98755 x 10^9 N · m^2/C^2) * (-1.6 x 10^-6 C) * (1.6 x 10^-6 C) / r
Rearranging the equation to solve for r:
r = (8.98755 x 10^9 N · m^2/C^2) * (-1.6 x 10^-6 C) * (1.6 x 10^-6 C) / -0.061 J
Calculating this expression:
r ≈ 0.687 meters
Therefore, the distance between the two spheres is approximately 0.687 meters.
To find the distance between the two spheres, we can use the formula for electrical potential energy and the Coulomb constant.
The formula for electrical potential energy is given by:
U = ((k * q1 * q2) / r)
where U is the electrical potential energy, k is the Coulomb constant, q1 and q2 are the charges on the two spheres, and r is the distance between the two spheres.
In this problem, one sphere loses electrons and the other sphere gains electrons. The charge on the first sphere would be positive, and the charge on the second sphere would be negative.
The charge on the first sphere (q1) can be found by multiplying the number of electrons removed by the charge on each electron:
q1 = (1 × 10^13) * (1.6 × 10^-19 C)
The charge on the second sphere (q2) would be the negative of q1.
Substituting these values into the formula for electrical potential energy, we get:
-0.061 J = ((8.98755 × 10^9 N · m^2/C^2) * (q1) * (-q1)) / r
Simplifying this equation, we can solve for the distance r:
r = sqrt((q1 * q1) / ((-0.061 J) * (8.98755 × 10^9 N · m^2/C^2)))
Now let's calculate the values and find the distance between the two spheres.
q1 = (1 × 10^13) * (1.6 × 10^-19 C) = 1.6 × 10^-6 C
r = sqrt((1.6 × 10^-6 C * 1.6 × 10^-6 C) / ((-0.061 J) * (8.98755 × 10^9 N · m^2/C^2)))
Calculating the above expression will give us the distance between the two spheres.
Please note that in this explanation, we have provided the step-by-step process to find the answer. You can use these steps with the given values to find the distance between the two spheres.