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tsuwal
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Homework Statement
Consider two conductors with capacity C1 and C2, separated by a distance d sufficiently large, so that each conductor can be considered as pontual, when observed by the other. Find the capacity coeficients and the capacity of the capacitor thereby formed.
Homework Equations
This problem is related to what my teacher calls the fundamental problem of eletrostatics:
- When given n conductors with know potencials, we want to calculate its charges. The charges of each conductor depend linearly on the potencials of the conductors such that:
Charge of conductor k=[itex]\sum_{i=1}^{n}S_{ik}V_i[/itex], where S_ik is the potencial coefficient and V_i is the potencial of conductor i
- When given n conductors with know charges, we want to calculate its potencials. The potencial of each conductor depend linearly on the charges of the conductors such that:
POtencial of conductor k=[itex]\sum_{i=1}^{n}C_{ik}q_i[/itex], where C_ik is the capacity coeffient and q_i is the charge of the conductor i
The Attempt at a Solution
How do I solve this? I tried this way:[itex]
\begin{bmatrix}
S_{11} &S_{12} \\
S_{21} & S_{22}
\end{bmatrix}
\begin{bmatrix}
V_1\\
V_2
\end{bmatrix}
=
\begin{bmatrix}
\frac{1}{C_1} & \frac{1}{4\pi\epsilon_0d}\\
\frac{1}{4\pi\epsilon_0d}& \frac{1}{C_2}
\end{bmatrix}
\begin{bmatrix}
V_1\\
V_2
\end{bmatrix}
=\begin{bmatrix}
q_1\\
q_2
\end{bmatrix}
[/itex]
I assumed S_11=1/C_1 and S_22=1/C_2 and S_12=S_12=1/4pi*epsilon*d
However, the answer I get from inverting this matrix is not the right answer. The right answer is:
[itex]C_{11}=\frac{(4\pi\epsilon_0d)^{2}C_1}{(4\pi\epsilon_0d)^{2}-C_1C_2};
C_{22}=\frac{(4\pi\epsilon_0d)^{2}C_2}{(4\pi\epsilon_0d)^{2}-C_1C_2};
C_{12}=C_{21}=-\frac{4\pi\epsilon_0dC_2C_1}{(4\pi\epsilon_0d)^{2}-C_1C_2}[/itex]
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