Help understanding the units in Gauss's law

[goldenwest]

Homework Statement

Well, here's the homework question:

A charge Q is located inside a rectangular box. The electric flux through each of the six surfaces of the box is: phi1=+1500Nm^2/C, phi2=+2200Nm^2/C, phi3=+4600Nm^2/C, phi4=-1800Nm^2/C, phi5=-3500Nm^2/C, and phi6=-3400Nm^2/C. What is Q?

My guess is that I will have to use Gauss's law (electric flux = enclosed charge over electric constant) to find out the charge for each number and then find the mean of all the charges? But what's really confusing me is that Gauss's law's units don't seem to cancel out properly. Please show me why I'm an idiot.

Homework Equations

seems like it should be

...why isn't it?

The Attempt at a Solution

My attempt at a solution would be
+1500 * 8.99E9 = 1.35E13
+2200 * 8.99E9 = 1.98E13
+4600 * 8.99E9 = 4.14E13
-1800 * 8.99E9 =-1.62E13
-3500 * 8.99E9 =-3.14E13
-3400 * 8.99E9 =-3.06E13

The mean of those numbers = -5.83E11 = Q?

 

[LowlyPion]

Welcome to PF.

Before puzzling over units maybe you need to get a clearer understanding of what needs doing?

http://hyperphysics.phy-astr.gsu.edu/hbase/electric/gaulaw.html#c2

 

[goldenwest]

Okay so $$(\sum\phi)\epsilon = Q$$? Q = -2.16E13?

 

[LowlyPion]

Okay so $$(\sum\phi)\epsilon = Q$$? Q = -2.16E13?

Are you sure that's the sum of the Φ ?

 

[goldenwest]

Oops, -3.60E12... but am I doing it correctly now?

 

[LowlyPion]

Oops, -3.60E12... but am I doing it correctly now?

So correctly that it looks like you got it right.

 

[goldenwest]

Okay, thank you very very very very much. I'm still confused about why the units work out that way - could you explain that to me?

 

[LowlyPion]

ε is in units of C²/Nm²

Φ is in units of Nm²/C

Q is in units of C

Edit: Sorry. I posted the units of Coulombs Constant which is the inverse.

 

[goldenwest]

Yes, I know. I know this is probably a stupid question... please bear with me.

$$\stackrel{Q}{\overline{\epsilon}} = \stackrel{C}{\overline{Nm^{2}C^{-2}}} = \stackrel{C^{3}}{\overline{Nm^{2}}}$$

Why is that wrong?

 

[LowlyPion]

Sorry. I posted the units of Coulombs Constant which is the reciprocal of the units of ε.

I corrected my error.

Electrical flux has SI units of volt metres (V m), or, equivalently, Newton metres squared per coulomb (N m² C−1).

http://en.wikipedia.org/wiki/Electric_flux

 

[goldenwest]

Edit: Sorry. I posted the units of Coulombs Constant which is the inverse.

Oooooooooook, now I understand. Thank you very much for all your help!

 

[goldenwest]

So was I using the wrong constant when solving for the problem... meaning I got the answer wrong? I was using 8.99E9. Should I be using 8.85E-12?

 

[LowlyPion]

So was I using the wrong constant when solving for the problem... meaning I got the answer wrong? I was using 8.99E9. Should I be using 8.85E-12?

Yes. I missed that too I guess. I should have checked it.

ε_o is what you want, and not k.

 

[goldenwest]

Thanks again, you have been extremely helpful.