26. Capacitors & Dielectrics

Find an expression for the capacitance of a spherical capacitor, consisting of concentric spherical shells of radii R₁ (inner shell) and R₂ (outer shell).

High-frequency signals are often transmitted along a coaxial cable, such as the one shown in FIGURE P26.68. For example, the cable TV hookup coming into your home is a coaxial cable. The signal is carried on a wire of radius R₁ while the outer conductor of radius R₂ is grounded (i.e., at V=0 V). An insulating material fills the space between them, and an insulating plastic coating goes around the outside. Evaluate the capacitance per meter of a cable having R₁=0.50 mm and R₂=3.0 mm.

A spherical capacitor contains a charge of $3.30$ nC when connected to a potential difference of $220$ V. If its plates are separated by vacuum and the inner radius of the outer shell is $4.00$ cm, calculate: (a) the capacitance; (b) the radius of the inner sphere; (c) the electric field just outside the surface of the inner sphere.

A capacitor is made from two hollow, coaxial, iron cylinders, one inside the other. The inner cylinder is negatively charged and the outer is positively charged; the magnitude of the charge on each is $10.0$ pC. The inner cylinder has radius $0.50$ mm, the outer one has radius $5.00$ mm, and the length of each cylinder is $18.0$ cm.

(a) What is the capacitance?

(b) What applied potential difference is necessary to produce these charges on the cylinders?

A spherical capacitor with a 1.0 mm gap between the shells has a capacitance of 100 pF. What are the diameters of the two spheres?

A coaxial cable, Fig. 24–35, consists of an inner cylindrical conducting wire of radius R_b surrounded by a dielectric insulator. Surrounding the dielectric insulator is an outer conductor of radius R_a, which is usually “grounded.” Determine an expression for the capacitance per unit length of a cable whose insulator has dielectric constant K.

A coaxial cable, Fig. 24–35, consists of an inner cylindrical conducting wire of radius R_b surrounded by a dielectric insulator. Surrounding the dielectric insulator is an outer conductor of radius Rₐ, which is usually “grounded.” For a given cable, R_b = 2.5 mm and R_a = 9.0 mm. The dielectric constant of the dielectric insulator is K = 3.2. Suppose that there is a potential of 1.0 kV between the inner conducting wire and the outer conducting sheath. Find the charge, and capacitance, per meter of the cable.