(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

A spherical conductor of radiusais surrounded by a spherical conducting shell of radiusb, and the gap is filled with an insulating material of resistivityρ. A thin wire connects the inner surface of the shell to the surface of the conductive sphere, and a potential ofVis applied to the outer surface of the conducting shell.

I.Determine the current drawn from the voltage source.

II.Integrate the power density (σE^{2}) over the insulator volumevand compare to the power drawn from the voltage source.

2. Relevant equations

dR = ρ dr / 4πr^{2}

i = V / R

E = ρ J = -[itex]\partial[/itex]V/[itex]\partial[/itex]r

P_{dissipative}= I^{2}R

P/v = σE^{2}

V = -∫Edot dr

3. The attempt at a solution

Part I was fairly straightforward. I found R by integrating from b to a (because the current travels inward from b) and got

R = ρ (b - a) / 4πab

which yielded

i = 4πVab / ρ(b - a)

Part II is giving me fits, however. The power drawn from the voltage source is just I*V. To get the power from the power density σE^{2}I would need to find an expression for E. SinceEis in the same direction as the radial displacement vectorr,

V = -∫E dr = -Er, evaluated over the limits of integration (frombtoa, in this case).

I have a feeling that this is the route I need to take, but I'm not sure where to take it from there. I also considered using

E = ρJ = ρ(i/A)

but am also not sure where to take that... I figure I would need to find a function J(r) since the current density is r-dependent. (Current is constant over an r-dependent geometry.)

Anyone who can help me with this, or give a hint as to the right direction, would be doing me a huge favor.

1. The problem statement, all variables and given/known data

2. Relevant equations

3. The attempt at a solution

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# Power draw of a spherical conductor surrounded by an insulator & a conducting shell

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