How Many Excess Electrons Cause a 4.57 * 10^-21 N Repulsive Force?

[stylez03]

Homework Statement

Two small spheres spaced 20.0 cm apart have equal charge.

How many excess electrons must be present on each sphere if the magnitude of the force of repulsion between them is 4.57 * 10^-21 N?

Homework Equations

F = 1/4pi*e_o * k * (|Q1|*|Q2| / r^2)

1/4pi*e_o = 8.988 * 10^-9

The Attempt at a Solution

Since we know the force, we just have to find how many excess electrons and since they have the same charge, we can just take that value and square it.

4.57 * 10^21 = 8.988 * 10^-9 * ((x * 10^-9)^2 / (.20)^2)

I've tried different values of X but I can't seem to get it to equate to given Force, and I'm not sure if I'm doing it right anymore.

 

[Kurdt]

If you manipulate the equation before putting the numbers in then it will be a lot easier.

$$q^2 =\frac{r^2F}{k}$$

The square root will give you the charge on each sphere and then you will have to divide that by the charge on an electron to determine the number of electrons on the sphere.

 

[stylez03]

q^2 = r^2*F / k

q = sqrt(r^2*F / k)

r = .20m
F = 4.57 * 10^-21
k = 8.85 * 10^-12

sqrt( ((.20)^2 * (4.57*10^-12)) / (8.85*10^-12) )

Doing the math I get q = .000005

You said to take this value and divide by the charge of an electron? I'm not sure what to do with this value after I obtain q, I know this should give you the charge of q right?

 

[Kurdt]

In the equation I gave you $$k=\frac{1}{4\pi \epsilon_0}$$ so the value you used is incorrect. As for q, that is the charge on 1 sphere and to find how many eectrons cause the charge you need to divide q by the charge on an electron.

 

[stylez03]

Yes, you're right I must of have missed that.

From the new calculations:

sqrt( ((.20)^2 * (4.57*10^-12)) / (1/ 4*pi*(8.85*10^-12)) )

q = 1.42*10^-16

Charge of Electron = 1.602 × 10^-19

So you said take q / charge of electron?

(1.42*10^-16) / (1.602* 10^-19) = 8.9*10^-36?

I entered that in as the solution, but the online program says it's still incorrect.

 

[Kurdt]

Should come out at about 890.

 

[stylez03]

Should come out at about 890.

Yea that's the answer, I guess I mis-interpreted the answer. Thanks!

 

[Kurdt]

Just looks like you made a minor error on the calculator when you were plugging the numbers in. If you try again you'll probably get the correct answer.