An air-filled spherical capacitor is constructed with inner...

An air-filled spherical capacitor is constructed with inner and outer shell radii of 7.00 and $14.0 \mathrm{cm},$ respectively. (a) Calculate the capacitance of the device. (b) What potential difference between the spheres results in a charge of $4.00 \mu \mathrm{C}$ on the capacitor?

Key Concepts

Relationship between Charge, Potential Difference, and Capacitance

This fundamental relationship, expressed as Q = C × V, connects the amount of charge stored (Q), the capacitance (C), and the potential difference (V) across a capacitor. It is a key principle in electrostatics because it allows us to determine one of these quantities if the other two are known. In problems involving capacitors, such as computing the voltage required to store a given charge, this relationship is central.

Spherical Capacitors

A spherical capacitor consists of two concentric spherical conductors with different radii, where one sphere is enclosed by the other. This geometry is used to store electrical energy by creating an electric field in the region between the two shells. The design and spatial separation of the conductors directly influence the capacitor's ability to store charge through the field established by the geometric configuration.

Capacitance

Capacitance is a measure of a capacitor's ability to store electric charge per unit potential difference between its conductors. For any capacitor, the capacitance is defined as C = Q/V, where Q is the amount of charge stored and V is the voltage difference. In the case of a spherical capacitor, the capacitance can be derived from the geometry of the spheres, showing a dependency on their radii and the separation between them.

Permittivity of Free Space

The permittivity of free space (denoted as ??) is a fundamental constant that describes how electric fields interact with the vacuum (or air, which behaves similarly). It appears in calculations of capacitance and plays a critical role in determining how much electric field is generated per unit charge. This constant is essential for linking the geometry of a capacitor to its ability to store electrical energy.

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