Locus of all points due to a charge

[tempneff]

Homework Statement

A 100 nC point charge is located at A(-1, 1, 3) in free space. (a) Find the locus of all point P(x,y,z) at which E_x = \frac{V}{m}. Find y_1 if P(-2,y_1,3) lies on that locus.

Homework Equations

E=k\frac{q}{r^2}\vec{r}\hspace{10pt}E_X=E\cos\theta\hspace{10pt}\vec{R}=<(x_2-x_1),(y_2-y_1),(z_2-z_1)>\hspace{10pt}
Cartesian to Spherical: r=\sqrt{x^2+y^2+z^2}\hspace{10pt}\theta = cos^{-1}\frac{z}{\sqrt{x^2+y^2+z^2}} \hspace{10pt} \phi = \cot^{-1}\frac{x}{y}

The Attempt at a Solution

I converted the vector R to spherical coordinates because I figured that using the radius I could find the field where \theta and \phi = 0 then convert back to cartesian...It didn't work. I had a hint that I should use E=k\frac{q}{\vert{\vec{R}\vert^3}}\vec{R} but I don't quite understand why. I believe this is just multiplying by 1, but I thought that while k\frac{q}{r^2}\vec{r} already include the vector I could just plug in \vec{R} for unit vector \vec{r} then why would I have to divide by the magnitude again.

I am sure at this point that I am misunderstanding things so any clarification of these concepts it priceless. Thanks in advance for any tips.

 

[haruspex]

E_x = \frac{V}{m}.

There seems to be a number missing. How many Volts/metre?
I see no merit in switching to polar. Use $E=k\frac{q}{r^2}\vec{r}$, replacing r with x, y, z. Given that you only want the x component, how will you replace $\vec{r}$?
(In the equation you posted with R³, the R vector appears to be the full vector, not the unit vector. Hence the need to divide by the magnitude again.)

 

[tempneff]

Right, that was supposed to be E_x=500\frac{V}{m}

I'm not sure how I can replace \vec{r} reflecting only the x direction. Also, if they are asking for a locus, do i have to come up with a item including x,y, and z rather than just x.

 

[haruspex]

I'm not sure how I can replace \vec{r} reflecting only the x direction.

The complete vector ($\vec R$?) is <x, y, z>. The unit vector is the same divided by its magnitude. So what is the x component of that?

Also, if they are asking for a locus, do i have to come up with a item including x,y, and z

An equation involving x, y and z, yes.