I would love to get a hint to do it any other way. Something that is more basic, like by using just the starting definition of C=Q/ V.hutchphd said:
You've been at PF long enough to know that any schoolwork-type problem like this needs to be posted in the Schoolwork forums, and you need to show your best efforts to work on the problem. This thread is now closed. Please re-post in the Schoolwork forums. Thank you.Kashmir said:
The general formula for capacitance between two points on three concentric shells is derived using the concept of series and parallel combinations of capacitors. If the shells have radii \( R_1 \), \( R_2 \), and \( R_3 \) with \( R_1 < R_2 < R_3 \), the capacitance between the innermost and outermost shells can be calculated by first finding the capacitance between each pair of shells and then combining them appropriately.
The capacitance between the innermost shell (radius \( R_1 \)) and the middle shell (radius \( R_2 \)) is given by the formula:\[ C_{1,2} = \frac{4 \pi \epsilon_0 R_1 R_2}{R_2 - R_1} \]where \( \epsilon_0 \) is the permittivity of free space.
The capacitance between the middle shell (radius \( R_2 \)) and the outermost shell (radius \( R_3 \)) is given by the formula:\[ C_{2,3} = \frac{4 \pi \epsilon_0 R_2 R_3}{R_3 - R_2} \]where \( \epsilon_0 \) is the permittivity of free space.
The total capacitance between the innermost and outermost shells, considering that the capacitances \( C_{1,2} \) and \( C_{2,3} \) are in series, is given by:\[ \frac{1}{C_{total}} = \frac{1}{C_{1,2}} + \frac{1}{C_{2,3}} \]Thus,\[ C_{total} = \left( \frac{1}{C_{1,2}} + \frac{1}{C_{2,3}} \right)^{-1} \]
Several assumptions are made in these calculations:1. The shells are perfect conductors.2. The space between the shells is filled with a vacuum or a uniform dielectric material.3. The shells are perfectly concentric.4. Edge effects are neglected, assuming that the shells are large enough for the edge effects to be insignificant.