Guass's Law over the x axis

1. Feb 19, 2013

Colts

1. The problem statement, all variables and given/known data
Charge is uniformly distributed along the x axis with density ß. Use Gauss' Law to find the electric field it produces, and use this to calculate the work done on a charge Q that moves along the y axis from y = a to y = b.

2. Relevant equations

$\phi$=$\int$$\vec{E}$*$\hat{n}$dA

$\phi$= $\frac{Q}{\epsilon}$
3. The attempt at a solution

I used a cylinder for my surface since the normal vector will always align with the electrical field. So the first part, the equation ends up
$\phi$=E$\int$dA
$\phi$=E(2∏rh)

(r is the radius from the axis to the edge of the cylinder and h is the length of the cylinder.)

and if I remember right, Q is the density times the are of enclosure.
so Q = β(2∏rh)
I set the two $\phi$ equations equal to each other and get
$\frac{β}{ε}$=E

I don't think that's right though. What did I do wrong?

Last edited: Feb 19, 2013
2. Feb 19, 2013

ap123

Since the charge is along the x-axis, then β is a linear charge density.
So, to get the charge enclosed, you multiply β by the length h, and not the volume.

3. Feb 19, 2013

tiny-tim

Hi Colts!
No, "density" here means the line density (in coulombs per metre, not per metre3)

So Q = βh.

(of course, sometimes "density" means surface density, and occasionally it actually means density! )

EDIT: ap123 beat me to it!

4. Feb 19, 2013

Colts

So the electrical field is

E = $\frac{β}{2πrε}$

Last edited: Feb 19, 2013
5. Feb 19, 2013

Colts

Is the last question asking me to integrate E from a to b?

6. Feb 19, 2013

SammyS

Staff Emeritus
Integrate the force, FExt, that would need to be exerted on a charge, Q, to move the charge from a to b . (Actually integrate the work the force does.)

7. Feb 19, 2013

Colts

$\int$$\frac{βxdx}{2πrε}$

x is the distance

does that look right? and the integral would be from a to b

8. Feb 19, 2013

SammyS

Staff Emeritus
Not correct.

What is the force on a charge Q located on the y-axis , a distance y from the x-axis ?

You (or some outside agent) must apply what force on Q to move it, at a constant rate, when the charge is located on the y-axis ?