A high-precision measurement device uses a long cylindrical capacitor consisting of four concentric metallic cylinders with radii 𝑅₁, 𝑅₂, 𝑅₃, and 𝑅₄. In this setup, all cylinders are made of copper. The cylinders Copper strips connect 2 and 3 to ensure they are at the same potential. Calculate the capacitance per unit length of this arrangement, assuming equal and opposite charges are placed on the innermost cylinder 𝑅₁ and the outermost cylinder 𝑅₄. The capacitance of the cylindrical capacitor is $C=\frac{2\pi \epsilon _0\ell }{\ln \left(\frac{R_{outer}}{R_{inner}}\right)}$.

• A. $\frac{4\pi \epsilon _0}{\ln \left(\frac{R_1R_3}{R_2R_4}\right)}$
• B. $\frac{4\pi \epsilon _0}{\ln \left(\frac{R_1R_4}{R_2R_3}\right)}$
• C. $\frac{2\pi \epsilon _0}{\ln \left(\frac{R_2R_4}{R_1R_3}\right)}$
• D. $\frac{2\pi \epsilon _0}{\ln \left(\frac{R_2R_3}{R_1R_4}\right)}$

Answer: (C)