 #1
L_ucifer
 12
 0
 Homework Statement:

A solid, nonconducting sphere of radius a has a volume charge density given by the equation ρ(r) = ρ0(r/a)^3, where r is the distance from the sphere’s centre.
(a) Determine the electric field magnitude, E(r), as a function of r.
(b) Determine the potential, V(r), as a function of r. Take the zero of potential at r = ∞.
 Relevant Equations:

E = dV/dr
V = W/q
I understand part (a) of this question, and my answer for that part is:
*For r < a*
E = (ρ_{0} * r^{4}) / (6 * ε_{0} * a^{3})
* For r ≥ a*
E = (ρ_{0} * a^{3}) / (6 * ε_{0} * r^{2})
Now, for part (b), I understand one solution is, for r < a, find the work done to bring a point charge q from infinity to a and then from a to r and divide the work by q. This solution is attached below:
Now, we also know that E = dV/dr. This checks out because when we take the negative derivative of V found in the solution, we get the equation for E when r < a. This derivative can also be written as: V = ∫E*dr; however, when I integrate the equation of E for r < a, I get a different solution. Specifically, my solution is the same as the given solution except it doesn't have 6 as a coefficient for a^{2}. Is there a gap in my understanding? What have I done wrong? I appreciate the support.
EDIT: Updated the variables and fixed the typesetting.
*For r < a*
E = (ρ_{0} * r^{4}) / (6 * ε_{0} * a^{3})
* For r ≥ a*
E = (ρ_{0} * a^{3}) / (6 * ε_{0} * r^{2})
Now, for part (b), I understand one solution is, for r < a, find the work done to bring a point charge q from infinity to a and then from a to r and divide the work by q. This solution is attached below:
Now, we also know that E = dV/dr. This checks out because when we take the negative derivative of V found in the solution, we get the equation for E when r < a. This derivative can also be written as: V = ∫E*dr; however, when I integrate the equation of E for r < a, I get a different solution. Specifically, my solution is the same as the given solution except it doesn't have 6 as a coefficient for a^{2}. Is there a gap in my understanding? What have I done wrong? I appreciate the support.
EDIT: Updated the variables and fixed the typesetting.
Last edited: