anonymous
  • anonymous
Three concentric spherical shells have radii a, b and c(a
Physics
jamiebookeater
  • jamiebookeater
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anonymous
  • anonymous
I think it's 3 but I can't seem to explain why.. It's a matter of reasoning going from the basis that the potentials are dependent on both radius and surface charge. Looking at the information given, there seems to be no way that any of the potentials could ever be equal to one of the others without disregarding one of the equations.
anonymous
  • anonymous
The answer in book is 1 but it can be a misprint as well though a small solution is also given in the book. Still it is beyond my understanding. I am giving the solution as given in book........\[Va=\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi a ^{2} }{ a }-\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi b ^{2} }{ b }+\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi c ^{2} }{ c }\] \[V _{b}=\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi a ^{2} }{ a }-\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi b ^{2} }{ b }+-\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi c ^{2} }{ c }\] \[V _{c}=\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi a ^{2} }{ a }-\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi b ^{2} }{ b }+\frac{ 1 }{ 4\pi \epsilon _{o} }\frac{ \sigma4\pi c ^{2} }{ c }\] Hence Va=Vb≠Vc
anonymous
  • anonymous
for c=a+b \(V_a=\frac{1}{4\pi \epsilon_0}( \frac{4\pi a^2 \sigma }{a}-\frac{4\pi b^2 \sigma }{b}+\frac{4\pi c^2 \sigma }{c})=\frac{\sigma}{\epsilon_0}(a-b+c)=\frac{\sigma}{\epsilon_0}(2a)\) \(V_b=\frac{1}{4\pi \epsilon_0}( \frac{4\pi a^2 \sigma }{b}-\frac{4\pi b^2 \sigma }{b}+\frac{4\pi c^2 \sigma }{c})=\frac{\sigma}{\epsilon_0}(\frac{a^2}{b}-b+c)=\frac{\sigma}{\epsilon_0}(\frac{a^2 +ab}{b})\) \(V_c=\frac{1}{4\pi \epsilon_0}( \frac{4\pi a^2 \sigma }{c}-\frac{4\pi b^2 \sigma }{c}+\frac{4\pi c^2 \sigma }{c})=\frac{\sigma}{\epsilon_0}(\frac{a^2-b^2}{a+b}+a+b)=\frac{\sigma}{\epsilon_0}(2a)\) thus answer is (1) do you need further help?

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anonymous
  • anonymous
note that the potential outside the charged sphere is \(V=\frac{r_0^2 \sigma}{\epsilon_0 R}\) but inside the sphere it's\(V=\frac{r_0 \sigma}{\epsilon_0 }\)

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