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Electric field energy density is the amount of energy stored per unit volume in an electric field. It is a critical concept when dealing with charged objects and understanding how energy is distributed in space around them. The formula to calculate the electric field energy density, denoted as \( u \), is:\[ u = \frac{1}{2} \varepsilon_0 E^2 \]where \( \varepsilon_0 \) is the permittivity of free space, and \( E \) is the electric field strength.

• Inside a conducting sphere with radius \( R \), the electric field is zero throughout the interior \(( r < R )\). Hence, the energy density inside is also zero.
• Outside the sphere, where \( r > R \), the electric field behaves as if all the charge \( Q \) is concentrated at the center of the sphere. The energy density can then be expressed as: \( u = \frac{1}{2} \varepsilon_0 \left( \frac{kQ}{r^2} \right)^2\).

This expression indicates how energy is dispersed variably in the space surrounding the charged sphere, declining with the square of the distance from the sphere.

The capacitance of an isolated sphere represents its ability to store electrical charge. It relates the charge \( Q \) on the sphere to the electric potential \( V \). The formula for capacitance \( C \) of a sphere with radius \( R \) is:\[ C = 4\pi\varepsilon_0 R \]To derive this, consider the potential \( V \) of the sphere, caused when charge \( Q \) is distributed across its surface. The potential can be represented as:\[ V = \frac{kQ}{R} \]Using the relationship \( C = \frac{Q}{V} \) and substituting for \( V \), we obtain:

• \( C = \frac{Q}{\frac{kQ}{R}} = \frac{Q}{\frac{Q}{4\pi\varepsilon_0 R}} \)
• This simplifies to \( C = 4\pi\varepsilon_0 R \), illustrating that the capacitance depends linearly on the sphere's radius.

Thus, the larger the sphere, the higher its capacitance, resulting in a greater ability to hold charge for a given potential.

To find the total electric energy stored in the electric field surrounding a conducting sphere, we integrate the energy density over the space outside the sphere. Using a spherical shell of radius \( r \) and thickness \( dr \), the incremental volume \( dV \) is:\[ dV = 4\pi r^2 dr \]The energy stored in this shell, \( dU \), is:\[ dU = u \cdot dV = \frac{1}{2} \varepsilon_0 \left( \frac{kQ}{r^2} \right)^2 \cdot 4\pi r^2 dr \]When integrated from the surface of the sphere \( R \) to infinity, we find the total energy \( U \):\[ U = \int_R^\infty \frac{k^2Q^2}{8\pi\varepsilon_0 r^2} \cdot 4\pi r^2 dr \]This simplifies and evaluates to:\[ U = \frac{kQ^2}{2R} \]This result signifies the total work done to assemble the charge \( Q \) onto the sphere, providing valuable insight into both theoretical and practical contexts of electric field applications. This integration also serves to illuminate how electric fields behave and interact over distance, influencing the distribution of forces and energy in space.