A capacitor is made from two hollow, coaxial, iron...

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 $\mathrm{pC}$ . The inner cylinder has radius $0.50 \mathrm{mm},$ the outer one has radius $5.00 \mathrm{mm},$ and the length of each cylinder is 18.0 $\mathrm{cm} .$ (a) What is the capacitance? (b) What applied potential difference is necessary to produce these charges on the cylinders?

Key Concepts

Permittivity of Free Space

The permittivity of free space is a constant that appears in many electromagnetic formulas, including those for capacitors. It relates the electric field in a vacuum to the charge that produces it and plays a significant role in determining the capacitance of various geometries, like the coaxial cylinders discussed in this context.

Electric Potential Difference

The electric potential difference represents the work done per unit charge in moving a charge between two points in an electric field. In capacitor problems, once the capacitance and stored charge are known, the potential difference can be directly calculated using the relation V = Q/C. It is a key quantity that connects the electrical properties of a capacitor to the energy stored within its electric field.

Coaxial Capacitor Geometry

When capacitors are made with cylindrical conductors arranged coaxially, the capacitance can be derived by analyzing the electric field between the cylinders. The geometry leads to a logarithmic dependence in the formula for capacitance, which incorporates the radii of the inner and outer conductors along with the permittivity of free space. This approach is widely used to study realistic capacitor designs such as those found in coaxial cables.

Capacitance

Capacitance is the measure of a system's ability to store electric charge per unit potential difference. It is defined by the equation Q = CV, where Q is the stored charge, C is the capacitance, and V is the voltage across the device. This concept is central to analyzing how much energy a capacitor can store and is a fundamental property in electrical circuits.

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