A 50.0 -m length of coaxial cable has an inner conductor...

A 50.0 -m length of coaxial cable has an inner conductor that has a diameter of 2.58 $\mathrm{mm}$ and carries a charge of $8.10 \mu \mathrm{C} .$ The surrounding conductor has an inner diameter of 7.27 $\mathrm{mm}$ and a charge of $-8.10 \mu \mathrm{C} .$ (a) What is the capacitance of this cable? (b) What is the potential difference between the two conductors? Assume the region between the conductors is air.

Key Concepts

Capacitance

Capacitance is a measure of a system's ability to store electric charge per unit of potential difference. It is a fundamental property of any system of conductors and is defined by the ratio of the magnitude of the charge stored on one conductor to the potential difference between the conductors.

Coaxial Cable Geometry

The geometry of a coaxial cable, featuring a central conductor and a surrounding conductor separated by a dielectric (such as air), plays a crucial role in determining its capacitance. The coaxial configuration leads to a capacitance per unit length that depends logarithmically on the ratio of the radii of the conductors, highlighting the importance of cylindrical geometry in electrical designs.

Electric Potential Difference

The electric potential difference, or voltage, between two conductors is the work done per unit charge in moving a test charge from one conductor to the other. This concept is central to understanding how stored energy is distributed in an electrical system and directly relates to the charge stored and the system's capacitance.

Permittivity of Free Space

The permittivity of free space (or that of the air in many practical cases) is a constant that appears in the equations for electric fields and capacitance. It quantifies the ability of a vacuum to permit electric field lines, and in designing capacitors, it determines how effectively the space between the conductors can support the electric field generated by the stored charge.

Charge-Capacitance-Voltage Relationship

This relationship, expressed as Q = CV, is a fundamental equation in electrostatics that links the amount of electric charge stored on a capacitor, its capacitance, and the potential difference across its terminals. It provides a direct method to calculate one parameter when the other two are known, and is vital for the analysis of capacitive systems.

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