A long cylindrical conductor has a length of 1 meter and is surrounded by a coaxial cylindrical conducting shell with an inner radius double that of the long cylindrical conductor. Assuming vacuum in the space between the cylinders, calculate the capacitance for this capacitor.
Given:
- Length of cylindrical conductor (\(\ell\)): 1 meter
- Inner radius of shell (\(a\)): \(a = \frac{1}{2}b\)
Formula:
\(C = \frac{2\pi\epsilon_0\ell}{\ln\left(\frac{b}{a}\right)}\)
Calculation:
\(C = \frac{2\pi\epsilon_0}{\ln(2)}\)
Substituting the values, we get:
\(C = \frac{2\pi \times 8.85 \times 10^{-12}}{\ln(2)} \approx 7.99 \times 10^{-11}\) Farads
Therefore, the capacitance of the capacitor is approximately \(7.99 \times 10^{-11}\) Farads.