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Electric potential between two concentric spherical shells

  1. Jun 4, 2017 #1
    1. The problem statement, all variables and given/known data
    (The complete problem statement and solution are inside the attached picture)
    Two isolated, concentric, conducting spherical shells have radii ##R_1=0.500 m## and ##R_2=1.00 m##, uniform charges ##q_1=2.00 mC## and ##q_2=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)##.

    2. Relevant equations
    $$V_f-V_i=-\int_i^f \vec E \cdot d\vec s\,$$
    or
    $$V=-\int_i^f \vec E \cdot d\vec s\,$$
    3. The attempt at a solution
    For part (f) and using the results of the previous parts of the problem:
    $$V(r)=-\int_{\infty}^{R_2} {E_1}(r) \,dr -\int_{R_2}^r {E_2}(r) \,dr=\frac {q_1 + q_2} {4 \pi \epsilon_0 r}+ \frac {q_1} {4 \pi \epsilon_0 r} - \frac {q_1} {4 \pi \epsilon_0 R_2}$$

    And it simplifies to:
    $$= \frac {1} {4 \pi \epsilon_0} (\frac {2q_1 + q_2} {r} - \frac {q_1} {R_2})$$
    Which is different than what's in the problem's official solution.
    Would someone please help me with this?
     

    Attached Files:

  2. jcsd
  3. Jun 4, 2017 #2

    TSny

    User Avatar
    Homework Helper
    Gold Member

    For the first term on the far right side, how do you get ##r## to appear in the denominator? Does this term come from the first integral? Note that this integral does not contain ##r## in the limits of integration.
     
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