Find the electric field intensity from an infinite line charge

[math132003]

Homework Statement

Consider that in an rectangular coordinate system an infinite charge line is placed exactly on the "x" axis. This line has a uniform charge distribution with linear charge density pL = 10 nC/m

(a) Determine the electric field intensity vector at point P = (4, 6, 8)

(b) What is the point charge value that should be placed at the point (0, -6, 8) so that the component of the Ey field at the same point P is null.

Relevant Equations

Electric field due to infinite line of charge, E = (pL/(2pi*r*p))*âp

what I've done so far?

-i've determined the vector between the point (4, 0, 0) and the point P.
(4, 6, 8) - (4, 0, 0)
(0, 6, 8)

-The norm of this vector is the radial distance of the line to point P (the value of “ρ” in the formula)
√(0^2 + 6^2 + 8^2) = 10 -> ρ = 10

-and its unit vector is the direction of the electric field (it is the unit vector âρ in the formula)
(1/10)*(0,6,8) -> (0, 3/5, 4/5)how can i continue?
Thanks in advance!

 

[BvU]

Hello @math132003. !

If I translate your notation to the hyperphysics notation: $$E = {\lambda\over 2\pi \,r\,\varepsilon_0}
$$pL $\qquad$ is $\qquad\lambda$
$\rho \qquad\$ is $\qquad$ r
p $\qquad$ is $\qquad\varepsilon_0$
And you use âp in the formula, but then write âρ in the comment

Confuses me somewhat -- please explain.

In the mean time I continue in the hyperphysics notation, adding a radial unit vector $\hat r$ in the $yz$ plane, so that
$$\vec E = {\lambda\over 2\pi \,r\,\varepsilon_0}\,\hat r
$$for which you have $r = 10$ and $\hat r = (0, 3/5,4/5)$.

So what's the problem ?

 

[math132003]

Hello @BvU!

Thanks for the translation and I apologize for the confusion. Langauge problems.

Even knowing this expression i still can't go ahead and develop these items:

(a) Determine the electric field intensity vector at point P = (4, 6, 8)

* here I just need to take everything I found and replace in the expression above? (b) What is the point charge value that should be placed at the point (0, -6, 8) so that the component of the Ey field at the same point P is null.

* I still don't know how to do anything on that item

Thank you again.

 

[BvU]

* here I just need to take everything I found and replace in the expression above?

Yes! And carefully check your answer (including units !)

(b) For this you need a formula for the electric field of a point charge Q. I don't spoil the exrcise if I tell you that the formula is$$\vec E = {Q\over 4\pi\,r^2\varepsilon_0}\hat r$$ where $\vec r$ is pointing from $(0,-6,8)$ to P. Your job to find the length and the $y$ component.
A sketch in the plane containing the $x$ axis and P might help.