Charge on Sphere: Solving Coulomb's Law - 12µC Total Charge

In summary, in order to find the positive charge on one of two identical small conducting spheres separated by 0.60 m and experiencing an attractive electric force of 10.8 N, the total charge of -12 micro C must be taken into account. Using Coulomb's Law and the rule for absolute values, it is determined that there are two possible solutions: one with a positive charge of 15.6 micro C and a negative charge of -27.6 micro C, and one with a positive charge of -27.6 micro C and a negative charge of 15.6 micro C.
  • #1
SheldonG
50
0

Homework Statement


Two identical small conducting spheres are separated by 0.60 m. The spheres carry different amounts of charges and each sphere experiences an attractive electric force of 10.8 N. The total charge on the two spheres is [itex]=-12\mu C[/itex]. What is the positive charge on one of the spheres?

Homework Equations


Coulomb's Law.

The Attempt at a Solution


Part of my problem is I think I am confused about how to treat the signs of the charges.

[tex] q_1 + q_2 = -12 \times 10^{-6} [/tex]
[tex] q_1 = -(q_2 + 12) [/tex]

Coulomb's Law gives:

[tex] 10.8 = k\frac{q_2(q2_+12)}{0.60^2} [/tex]

I left out the negative sign, because of the absolute value in Coulomb's law.
Is this correct?

and leads me to

[tex] 0 = q_2^2 + (12\times 10^{-6})q_2 - 4.32601 \times 10^{-10} [/tex]

From this, I end up with two solutions that fit the conditions of the problem. Is that right? I get a positive root of 27.6 micro C, and a negative root of 15.6 micro C, which gives me the positive charge of 3.6 micro C.

Could anyone confirm that this is the right way to handle this situation, or clear up my sign confusion for me?

Thanks so much,
Sheldon
 
Last edited:
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  • #2
Indeed, the signs seems to be your problem, the first line should read

[tex] q_1 + q_2 = 12 \times 10^{-6} [/tex]
 
  • #3
Hi Hootenanny,

Sorry, I made a typo when I entered the problem. The sum of the charges is -12 micro C. I'll edit the original post to fix that.

Sheldon
 
  • #4
Then your method looks good to me...:approve:
 
  • #5
Thanks! So there are actually two possible pairs of charges for this situation? Interesting...
 
  • #6
SheldonG said:

Homework Statement


Two identical small conducting spheres are separated by 0.60 m. The spheres carry different amounts of charges and each sphere experiences an attractive electric force of 10.8 N. The total charge on the two spheres is [itex]=-12\mu C[/itex]. What is the positive charge on one of the spheres?


Homework Equations


Coulomb's Law.


The Attempt at a Solution


Part of my problem is I think I am confused about how to treat the signs of the charges.

[tex] q_1 + q_2 = -12 \times 10^{-6} [/tex]
[tex] q_1 = -(q_2 + 12) [/tex]

Coulomb's Law gives:

[tex] 10.8 = k\frac{q_2(q2_+12)}{0.60^2} [/tex]

I left out the negative sign, because of the absolute value in Coulomb's law.
Is this correct?

and leads me to

[tex] 0 = q_2^2 + (12\times 10^{-6})q_2 - 4.32601 \times 10^{-10} [/tex]

From this, I end up with two solutions that fit the conditions of the problem. Is that right? I get a positive root of 27.6 micro C, and a negative root of 15.6 micro C, which gives me the positive charge of 3.6 micro C.

Could anyone confirm that this is the right way to handle this situation, or clear up my sign confusion for me?

Thanks so much,
Sheldon

whatch out.. there must be something wrong (I will check in a second) because it's clear that with your two solutions, the product of the charges is not the same so both can't be solutions (I mean that 27.6 times (-39.6) is obviously not the same as -15.6 times +3.6) !
 
  • #7
SheldonG said:

Homework Statement


Two identical small conducting spheres are separated by 0.60 m. The spheres carry different amounts of charges and each sphere experiences an attractive electric force of 10.8 N. The total charge on the two spheres is [itex]=-12\mu C[/itex]. What is the positive charge on one of the spheres?


Homework Equations


Coulomb's Law.


The Attempt at a Solution


Part of my problem is I think I am confused about how to treat the signs of the charges.

[tex] q_1 + q_2 = -12 \times 10^{-6} [/tex]
[tex] q_1 = -(q_2 + 12) [/tex]

Coulomb's Law gives:

[tex] 10.8 = k\frac{q_2(q2_+12)}{0.60^2} [/tex]

I left out the negative sign, because of the absolute value in Coulomb's law.
Is this correct?
[/itex]
You have to be careful.

What they give is the magnitude, so you should really write
[tex] 10.8 = k \frac{|q_2| \times |q_2 +12|}{0.6^2} [/tex]

Now, what to do with the absolute values? the rule is

[tex] |q_2| = q_2 ~if~ q_2 >0 [/tex]
[tex] |q_2| = -q_2 ~if~ q_2 <0 [/tex]
[tex] |q_2+12| = q_2 +12~if~ q_2+12 >0 \rightarrow q_2> -12 [/tex]
[tex] |q_2| =- q_2 -12~if~ q_2 +12<0 \rightarrow q_2 < -12[/tex]


So there are several cases to consider. If q_2 > 0, then you may use [itex] |q_2| |q_2 +12| = q_2 (q_2+12)[/itex] like you did. this means that you must reject any solution with q_2 negative from your solution.
The solution of that quadratic are [itex] -27.6 \muC [/itex] or [itex]+15.6 \muC
[/itex] Then we keep only the +15.6 solution which means the other charge is -27.6 microcoulombs (this gives an attractive force as the question asks and this gives the correc magnitude of the force as can be checked.



Then you may consider -12< q_2 < 0 in which case [itex] |q_2| |q_2 +12| = - q_2 (q_2+12)[/itex] This gives you a different quadratic solution that you will solve and accept the solution only if q_2 is in the correct range. It turns out there is no solution.


Finally, if q_2 < -12, you have to use [itex] |q_2| |q_2 +12| = -q_2\times (-q_2-12)[/itex] which will give you back the same equation you had so the solutions will be the same but now you reject any solution that is not below -12. This leaves the solution of -27.6 microC so the other charge is +15.6 microC.

This makes sense...there are two solutions in the sense that either charge may be -27.6 microC with the other being +15.6 microC.


Patrick
 
  • #8
Patrick,

Thank you. That was incredibly enlightening. I really appreciate it!

Sheldon
 

1. What is Coulomb's Law?

Coulomb's Law is a fundamental law of electrostatics that describes the force between two charged particles. It states that the force is directly proportional to the product of the charges and inversely proportional to the square of the distance between them.

2. What is the charge on a sphere?

The charge on a sphere refers to the total amount of charge contained within the surface of a sphere. It can be positive or negative, depending on the type of charge present.

3. How do you solve for the charge on a sphere using Coulomb's Law?

To solve for the charge on a sphere using Coulomb's Law, you will need to know the magnitude of the force acting on the sphere, the distance between the sphere and the charged particle, and the permittivity of the surrounding medium. Using these values, you can rearrange the equation to solve for the charge on the sphere.

4. What is the unit of charge in Coulomb's Law?

The unit of charge in Coulomb's Law is Coulomb (C), which is equivalent to an Ampere-second (A*s).

5. How can the charge on a sphere affect the surrounding electric field?

The charge on a sphere can affect the surrounding electric field by creating an electric potential that can attract or repel other charged particles. The strength of the electric field is directly proportional to the charge on the sphere and inversely proportional to the square of the distance from the sphere.

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