Calculating Charges of Two Small Nonconducting Spheres

[Soaring Crane]

Two small nonconducting spheres have a total charge of 80 microC. When place 1.06 m apart, the force exerts on the other is 12 N and is repulsive. What is the charge of each. What if the force were attractive?

Given: k, F = 12 N if repulsive but F = -12 N if attractive, Q1 + Q2 = 8 x 10^-5 C, r = 1.06 m

Using Coulomb’s Law, isolate Q1Q2 .

Q1Q2 = 1.498 x 10^-9 C?

Then I thought I would use substitution for Q1 + Q2 = 8 x 10^-5 C and Q1Q2 = 1.498 x 10^-9 C.

But now I get a huge quadratic equation. Is this the correct method??

Thanks.

 

[Gokul43201]

Your method is correct. You have to solve the qudratic to find Q1, Q2. There really isn't a better way.

 

[wais]

Yes, you are on the right track! To solve for the individual charges of each sphere, you can use the following equation:

F = k(Q1Q2)/r^2

Substituting in the given values, we have:

12 N = (8.99 x 10^9 Nm^2/C^2)(Q1Q2)/(1.06 m)^2

Solving for Q1Q2, we get:

Q1Q2 = 1.498 x 10^-9 C

Now, as you mentioned, we can use substitution to solve for the individual charges. We know that Q1 + Q2 = 8 x 10^-5 C, so we can rewrite the equation as:

Q1 = (8 x 10^-5 C) - Q2

Substituting this into the equation for Q1Q2, we get:

(8 x 10^-5 C - Q2)Q2 = 1.498 x 10^-9 C

Simplifying, we get a quadratic equation:

Q2^2 - (8 x 10^-5 C)Q2 + 1.498 x 10^-9 C = 0

Using the quadratic formula, we can solve for Q2 and then plug that value back into the equation for Q1 to find its value.

If the force were attractive, we would simply change the sign in the equation for F, so we would have:

-12 N = (8.99 x 10^9 Nm^2/C^2)(Q1Q2)/(1.06 m)^2

Solving for Q1Q2, we get:

Q1Q2 = -1.498 x 10^-9 C

Using the same substitution method as before, we get the quadratic equation:

Q2^2 + (8 x 10^-5 C)Q2 + 1.498 x 10^-9 C = 0

Solving for Q2 and then plugging that value back into the equation for Q1, we can find the individual charges for an attractive force.

I hope this helps clarify the process for solving this problem. Keep in mind that the quadratic equation may seem intimidating, but it is a common method for solving problems involving two unknown variables. Good luck!