Electric Field & Potential from Two Conductors

[Potatochip911]

Homework Statement

Two isolated, concentric, conducting spherical shells have radii R1=0.500 m and R2=1.00 m, uniform charges q1=2.00 mC and q2=1.00 mC, and negligible thicknesses. What is the magnitude of the electric field E at radial distance (a) r=4.00 m, (b) r=0.700 m, and (c) r=0.200 m? With V=0 at infinity, what is V at (d) r=4.00 m, (e) r=1.00 m, (f) r=0.700 m, (g) r=0.500 m,(h) r=0.200 m, and (i) r=0? ( j) Sketch E(r) and V(r).

Homework Equations

##\oint \vec{E}\cdot\vec{dA}=\frac{q_{encl}}{\varepsilon_0}##
##V_f-V_i=-\int_i^f\vec{E}\cdot d\vec{s}##

The Attempt at a Solution

I'm not having any trouble with the electric fields so I will just list the results:

##r<R_1, \hspace{5mm} E=0##
##R_1<r<R_2, \hspace{5mm} E=\frac{q_1}{4\pi\varepsilon_0 r^2}##
##R_1<R_2<r, \hspace{5mm} E=\frac{q_1+q_2}{4\pi\varepsilon_0 r^2}##

I am quite confused by the electric potential however, taking ##V_i=0## at infinity, I don't understand the process of calculating the electric potential.

$$V=-\int_{\infty}^{r}\vec{E}\cdot d\vec{s}$$

I'm not sure what to replace ##d\vec{s}## with, perhaps a vector ##d\vec{r}## that goes outwards radially and then integrating along the radius. Although, suppose I want to integrate to an ##r>R_2##, the Electric Field changes throughout the the integration at points ##r=R_1## and ##r=R_2##. How would I go about doing this?

Edit: Solved, split integral up so that the correct electric field is in place for the desired radius

 

[Simon Bridge]

Well done.