Electricity, Electic Force, and Charge Transfer

[Kaoi]

Electricity, Electric Force, and Charge Transfer

Homework Statement

"Two conducting spheres have identical radii. Initially, they have charges of opposite sign and unequal magnitudes with the magnitude of the positive charge larger than the magnitude of the negative charge. They attract each other with a force of 0.224 N when separated by 0.4 m.

The spheres are suddenly connected with a thin wire, which is then removed. Now the spheres repel each other with a force of 0.039 N. What are the magnitudes of the initial positive and negative charges? Answer in units of C."

Given:
$$\bullet F_{e,i} = -0.224 N$$
$$\bullet r = 0.4 m$$
$$\bullet F_{e,f} = 0.039 N$$
$$\bullet k_{C} = 8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$$

Unknown:
$$\bullet q_{pos} = ?$$
$$\bullet q_{neg} = ?$$

Homework Equations

$$F_{e} = \frac{k_{C}q_{1}q_{2}}{r^2}$$

The Attempt at a Solution

$$q_{pos} + q_{neg} = 2 q_{f}$$

$$F_{e,i} = \frac{k_{C}q_{pos}q_{neg}}{r^2}$$

$$F_{e,f} = \frac{k_{C}q_{f}^2}{r^2}$$

$$q_{f} = \sqrt{\frac{F_{e,f}r^2}{k_{C}}} = \frac {q_{pos} + q_{neg}}{2}$$

$$q_{pos}+q_{neg} = 2r\sqrt{\frac{F_{e,f}}{k_{C}}$$

$$q_{pos} = 2r\sqrt{\frac{F_{e,f}}{k_{C}}}- q_{neg}$$

At this point, I wasn't sure how to go about finding either of these charges, because you need one to solve for the other. Is there some clever mathematical trick or physical concept I'm missing here?

 

[Kurdt]

From the original force equation (i.e. |F|=0.224) you can substitute for one value of the charge and obtain an equation with just a single unknown. Also this might need looking at:

$$q_{pos} + q_{neg} = 2 q_{f}$$

 

[Kaoi]

From the original force equation (i.e. |F|=0.224) you can substitute for one value of the charge and obtain an equation with just a single unknown.

If I solve that, I get:

$$F_{e,i} = \frac{k_{C}q_{pos}{q_{neg}}}{r^2}$$

$$q_{pos}q_{neg} = \frac{F_{e,i}r^2}{k_{C}}$$

And I can't solve for $$q_{pos}$$ and $$q_{neg}$$ at the same time...
I'm not sure I understand what you mean by "substitution" here...

Also this might need looking at:

$$q_{pos} + q_{neg} = 2 q_{f}$$

Wouldn't the negative charge cancel out part of the positive charge?

$$q_{pos} + q_{neg} = q_{pos,remaining}$$

And then, since all that's left is positive charge, wouldn't the protons repel each other and spread evenly through the two connected spheres?

$$q_{f} = \frac{1}{2}q_{pos, remaining}$$

So:

$$2q_{f} = q_{pos,remaining} = q_{pos} + q_{neg}$$

At least, that's how I see it.

 

[Kurdt]

If I solve that, I get:

$$F_{e,i} = \frac{k_{C}q_{pos}{q_{neg}}}{r^2}$$

$$q_{pos}q_{neg} = \frac{F_{e,i}r^2}{k_{C}}$$

And I can't solve for $$q_{pos}$$ and $$q_{neg}$$ at the same time...
I'm not sure I understand what you mean by "substitution" here...

Try

$$q_{pos} = \frac{F_{e,i}r^2}{k_{C}q_{neg}}$$

Then substitute into:

$$q_{pos} = 2r\sqrt{\frac{F_{e,f}}{k_{C}}}- q_{neg}$$

and see how you go from there.

 

[Kaoi]

$$\frac{F_{e,i}r^2}{k_{C}q_{neg}} = 2r\sqrt{\frac{F_{e,f}}{k_{C}}} - q_{neg}$$

$$q_{neg}+\frac{F_{e,i}r^2}{k_{C}q_{neg}} = 2r\sqrt{\frac{F_{e,f}}{k_{C}}}$$

$$q_{neg}^2 + \frac{F_{e,i}r^2}{k_{C}} = 2rq_{neg}\sqrt{\frac{F_{e,f}}{k_{C}}}$$

$$q_{neg}^2 - 2rq_{neg}\sqrt{\frac{F_{e,f}}{k_{C}}} + \frac{F_{e,i}r^2}{k_{C}} = 0$$

So do I just solve that with the quadratic equation? (Sounds troublesome.)

 

[Kurdt]

I don't like this:

$$q_{pos} + q_{neg} = 2 q_{f}$$

You must remember sign conventions. If you fiddle with the signs you get a quadratic where you can complete the square.

 

[Kaoi]

I don't like this:

$$q_{pos} + q_{neg} = 2 q_{f}$$

You must remember sign conventions. If you fiddle with the signs you get a quadratic where you can complete the square.

But if the second charge is negative, wouldn't subtracting it make it addition?

 

[Kurdt]

All I was saying is you are adding a negative charge thus,

$$q_p+(-q_n) = q_p - q_n$$

Then your final equation will be solvable by the method of completing the square.

 

[Kaoi]

Changing the signs, I get basically the same equation with a couple of signs reversed-- that still doesn't help me much...

$$q_{neg}^2 + 2rq_{neg}\sqrt{\frac{F_{e,f}}{k_{C}}} - \frac{F_{e,i}r^2}{k_{C}} = 0$$

That's a strange square to complete...

 

[Kurdt]

Have you ever heard of the completing the square method?

$$(x+a)^2 = x^2+2ax+a^2$$
$$x^2+2ax= (x+a)^2-a^2$$

Does the equation you have look familiar?

 

[Kaoi]

So:

$$q_{neg}^2 + 2rq_{neg}\sqrt{\frac{F_{e,f}}{k_{C}}} - \frac{F_{e,i}r^2}{k_{C}} = 0$$

$$q_{neg}^2 + 2r\sqrt{\frac{F_{e,f}}{k_{C}}}q_{neg} = \frac{F_{e,i}r^2}{k_{C}}$$

$$x=q_{neg}$$, I know that, which would make:

$$q_{neg}^2 + 2q_{f}q_{neg} = (q_{neg} + q_{f})^2 - q_{f}^2$$

and that almost works, but $$F_{e,i}$$ and $$F_{e,f}$$ are two different quantities, so I can't use $$\sqrt{\frac{F_{e}r^2}{k_{C}}} = q_{f} = a$$, right? It looks like that's the only thing getting in the way. (This got a lot more complicated than I thought it would be...)

 

[Kurdt]

You can use $$\sqrt{\frac{F_{e}r^2}{k_{C}}}$$ as a. Just apply the above relation that I gave. The other term in the equation just tags on the end unchanged.

 

[Kaoi]

But the problem lies in which $$F_{e}$$ to use, since there are different $$F_{e}$$s on the sides of the equation, and they're not equal.

 

[Kurdt]

If we say $$\sqrt{\frac{F_{e}r^2}{k_{C}}}= k$$

Then,

$$q_{neg}^2 + 2q_{neg}k - \frac{F_{e,i}r^2}{k_{C}} = 0$$

$$(q_{neg}+k)^2 -k^2 - \frac{F_{e,i}r^2}{k_{C}} = 0$$

$$\left(q_{neg}+\sqrt{\frac{F_{e}r^2}{k_{C}}}\right)^2 - \frac{F_{e}r^2}{k_{C}} - \frac{F_{e,i}r^2}{k_{C}} = 0$$

 

[Kaoi]

That clears it up a lot for me. Sorry for making you have to go to so much trouble.

So:

$$\left(q_{neg}+\sqrt{\frac{F_{e,f}r^2}{k_{C}}}\right) ^2 - \frac{F_{e,f}r^2}{k_{C}} - \frac{F_{e,i}r^2}{k_{C}} = 0$$

$$q_{neg} + \sqrt{\frac{F_{e,f}r^2}{k_{C}}} = \sqrt{\frac{F_{e,f}r^2}{k_{C}} + \frac{F_{e,i}r^2}{k_{C}}}$$

$$q_{neg} = \sqrt{\frac{F_{e,f}r^2 + F_{e,i}r^2}{k_{C}}} - \sqrt{\frac{F_{e,f}r^2}{k_{C}}}$$

And then it's just simple addition.

Physics is hard.

 

[Kurdt]

That clears it up a lot for me. Sorry for making you have to go to so much trouble.

So:

$$\left(q_{neg}+\sqrt{\frac{F_{e,f}r^2}{k_{C}}}\right) ^2 - \frac{F_{e,f}r^2}{k_{C}} - \frac{F_{e,i}r^2}{k_{C}} = 0$$

$$q_{neg} + \sqrt{\frac{F_{e,f}r^2}{k_{C}}} = \sqrt{\frac{F_{e,f}r^2}{k_{C}} + \frac{F_{e,i}r^2}{k_{C}}}$$

$$q_{neg} = \sqrt{\frac{F_{e,f}r^2 + F_{e,i}r^2}{k_{C}}} - \sqrt{\frac{F_{e,f}r^2}{k_{C}}}$$

And then it's just simple addition.

Physics is hard.

At least getting the other charge is not as hard. Physics ain't so bad just need to practise those basics such as quadratic equations and basic calculus etc. Looks like your algebra and manipulation is top notch though.