Show that the electrostatic energy stored in the electric...

Show that the electrostatic energy stored in the electric field outside an isolated spherical conductor of radius $r_{0}$ carrying a net charge $Q$ is $$ U=\frac{1}{8 \pi \epsilon_{0}} \frac{Q^{2}}{r_{0}} $$ Do this in three ways: $(a)$ Use Eq. $24-6$ for the energy density in an electric field [Hint: Consider spherical shells of thickness $d r] ;(b)$ use Eq. $24-5$ together with the capacitance of an isolated sphere (Section $24-2$ ); (c) by calculating the work needed to bring all the charge $Q$ up from infinity in infinitesimal bits $d q$.

Key Concepts

Integration in Spherical Coordinates

For systems exhibiting spherical symmetry, such as an isolated spherical conductor, employing spherical coordinates simplifies the integration process. By considering spherical shells of varying radii and appropriate volume elements, one can efficiently integrate quantities like energy density over an entire spherical region, yielding the total energy stored in the field.

Capacitance and Energy Storage

Capacitance is a measure of a conductor's ability to store charge per unit potential difference. The energy stored in a system can be related to its capacitance through the equation U = Q²/(2C). This relationship provides a direct way to calculate electrostatic energy based on the geometry of the conductor and its capacitance, linking physical configuration to energy storage.

Electrostatic Energy Density

This concept defines the energy stored per unit volume in an electric field, typically expressed as u = (1/2)??E² in SI units. It enables the calculation of the total energy by integrating the energy density over the region in which the electric field exists, and is fundamental in connecting field properties with stored energy.

Work Done in Assembling Charge

This approach involves calculating the incremental work required to assemble a charge configuration, by bringing infinitesimal amounts of charge from infinity against an electric potential. Integrating the work done by these incremental additions gives the total energy stored in the configuration, offering a method that directly connects the process of charge assembly to energy accumulation.

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