3. A cylindrical capacitor consists of a cylinder of radius R, surrounded by a coaxial cylinder shell of inner radius R, (see Fig.3 below, R, > R,). Both cylinders have length L which we assume is much greater than the separation of the cylinders R, – R, , so that we can neglect any end effects. Now the capacitor is charged (by connecting it to a battery) so that the inner cylinder carries a charge +Q and the other one a charge -Q. Note that charges are only distributed over the outer surface of the inner cylinder and the inner surface of the outer cylinder, because of electrostatic equilibrium for these metal conductors; as a result, the electrostatic field only exists in the empty space between two cylinders. Determine: (a). The electric field E(r) as a function of the radius r measured with respect to the central axis of the cylinders. [hint: draw a proper Gaussian surface and then use Gauss's law.] (b). The potential difference (voltage) V between two cylinders. (c). The capacitance C of this cylindrical capacitor. (d). The electric energy U stored in the capacitor. 0- Rp Rp (a) (b) Fig.3 for Problem 3. (a). Cylindrical capacitor consists of two coaxial cylinder conductors. (b). The electric field lines are shown in cross-sectional view.

3. A cylindrical capacitor consists of a cylinder of radius R, surrounded by a coaxial cylinder shell of inner radius R, (see Fig.3 below, R, > R,). Both cylinders have length L which we assume is much greater than the separation of the cylinders R, – R, , so that we can neglect any end effects. Now the capacitor is charged (by connecting it to a battery) so that the inner cylinder carries a charge +Q and the other one a charge -Q. Note that charges are only distributed over the outer surface of the inner cylinder and the inner surface of the outer cylinder, because of electrostatic equilibrium for these metal conductors; as a result, the electrostatic field only exists in the empty space between two cylinders. Determine: (a). The electric field E(r) as a function of the radius r measured with respect to the central axis of the cylinders. [hint: draw a proper Gaussian surface and then use Gauss's law.] (b). The potential difference (voltage) V between two cylinders. (c). The capacitance C of this cylindrical capacitor. (d). The electric energy U stored in the capacitor. 0- Rp Rp (a) (b) Fig.3 for Problem 3. (a). Cylindrical capacitor consists of two coaxial cylinder conductors. (b). The electric field lines are shown in cross-sectional view.

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