- #1
LeePhilip01
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Hi I'm sorry if this is posted in the wrong section or it's laid out wrong but I have a question that I need a bit of help with.
I'm given: A spherically symmetrical charge distribution is contained within a sphere of radius a with no charge outside. At a distance r (r [itex]\leq[/itex] a) from the centre of the sphere the potential may be written as:
V = -Ar[itex]^{3}[/itex] + B
Then I'm asked to calculate a bunch of things which hopefully I've got right.
(a) Calculate the electric field intensity for r < a.
(b) What is the electric field intensity for r = a.
(c) Using Gauss's law calculate the total charge in the sphere.
(d) What is the electric field for r > a?
(e) What is the potential for r > a if V[itex]_{\infty}[/itex] = 0?
(f) Show that the difference in potential between r = a/2 and r = 2a is given by (19/8)Aa[itex]^{3}[/itex].
It's the last one I'm stuck on.
Gauss's Law.
E = -[itex]\nabla[/itex] V
Ok. So for (a) I differentiated and got E = 3Ar[itex]^{2}[/itex]
(b) I just subbed r=a and got E = 3Aa[itex]^{2}[/itex]
(c) I got that E for r=a with Gauss's Law: E = [itex]\frac{Q}{4\pi\epsilon_{0}a^{2}}[/itex]. Then equated that with answer from (b) to get Q = 12[itex]\pi\epsilon_{0}[/itex]Aa[itex]^{4}[/itex]
(d) I used Gauss's Law again to obtain: E = [itex]\frac{Q}{4\pi\epsilon_{0}r^{2}}[/itex].
Then I subbed in Q from part (c) to get: E = 3A[itex]\frac{a^{4}}{r^{2}}[/itex].
(e) I integrated -E with boundaries r and 0 and got: V = 3A[itex]\frac{a^{4}}{r}[/itex].
(f) Now I subbed r=a/2 into the equation for V given in the question. And r=2a into the equation I just got in part (e). But cannot seem to get (19/8)Aa[itex]^{3}[/itex] at all.
Thanks for any help.
Homework Statement
I'm given: A spherically symmetrical charge distribution is contained within a sphere of radius a with no charge outside. At a distance r (r [itex]\leq[/itex] a) from the centre of the sphere the potential may be written as:
V = -Ar[itex]^{3}[/itex] + B
Then I'm asked to calculate a bunch of things which hopefully I've got right.
(a) Calculate the electric field intensity for r < a.
(b) What is the electric field intensity for r = a.
(c) Using Gauss's law calculate the total charge in the sphere.
(d) What is the electric field for r > a?
(e) What is the potential for r > a if V[itex]_{\infty}[/itex] = 0?
(f) Show that the difference in potential between r = a/2 and r = 2a is given by (19/8)Aa[itex]^{3}[/itex].
It's the last one I'm stuck on.
Homework Equations
Gauss's Law.
E = -[itex]\nabla[/itex] V
The Attempt at a Solution
Ok. So for (a) I differentiated and got E = 3Ar[itex]^{2}[/itex]
(b) I just subbed r=a and got E = 3Aa[itex]^{2}[/itex]
(c) I got that E for r=a with Gauss's Law: E = [itex]\frac{Q}{4\pi\epsilon_{0}a^{2}}[/itex]. Then equated that with answer from (b) to get Q = 12[itex]\pi\epsilon_{0}[/itex]Aa[itex]^{4}[/itex]
(d) I used Gauss's Law again to obtain: E = [itex]\frac{Q}{4\pi\epsilon_{0}r^{2}}[/itex].
Then I subbed in Q from part (c) to get: E = 3A[itex]\frac{a^{4}}{r^{2}}[/itex].
(e) I integrated -E with boundaries r and 0 and got: V = 3A[itex]\frac{a^{4}}{r}[/itex].
(f) Now I subbed r=a/2 into the equation for V given in the question. And r=2a into the equation I just got in part (e). But cannot seem to get (19/8)Aa[itex]^{3}[/itex] at all.
Thanks for any help.