A capacitor is formed from two concentric spherical...

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius $12.5 \mathrm{cm},$ and the outer sphere has radius 14.8 $\mathrm{cm}$ . A potential difference of 120 $\mathrm{V}$ is applied to the capacitor. (a) What is the energy density at $r=12.6 \mathrm{cm},$ just outside the inner sphere? (b) What is the energy density at $r=14.7 \mathrm{cm},$ just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

A capacitor is formed from two concentric spherical conducting shells separated by vacuum. The inner sphere has radius $12.5 \mathrm{cm},$ and the outer sphere has radius 14.8 $\mathrm{cm}$ . A potential difference of 120 $\mathrm{V}$ is applied to the capacitor. (a) What is the energy
density at $r=12.6 \mathrm{cm},$ just outside the inner sphere? (b) What is the energy density at $r=14.7 \mathrm{cm},$ just inside the outer sphere? (c) For a parallel-plate capacitor the energy density is uniform in the region between the plates, except near the edges of the plates. Is this also true for a spherical capacitor?

Key Concepts

Spherical Capacitor

A spherical capacitor is a device consisting of two concentric spherical conductive shells separated by a dielectric (in this case, vacuum). This setup creates a radial electric field that varies with distance from the center. Understanding spherical capacitors is essential in electrostatics because their geometry leads to a distinct distribution of electric fields and potential differences compared to other capacitor geometries.

Electric Field in Spherical Geometry

In spherical systems, the electric field emanating from a charged conductor decreases with the square of the distance from the center, a result that can be derived using Gauss’s law. The radial symmetry simplifies the analysis, as the field at any point depends solely on the distance from the center. This concept is crucial for determining quantities such as potential differences and energy densities at various radial positions within the capacitor.

Energy Density

Energy density in an electric field quantifies the energy stored per unit volume and is given by the formula u = 1/2 ?0 E^2 in vacuum. This concept is key in electromagnetism because it allows the analysis of how energy is distributed in space, and how modifications in geometry or field strength directly affect the energy stored in capacitive systems.

Gauss's Law

Gauss's law is a fundamental principle of electrostatics used to relate the electric field to the charge enclosed by a closed surface. For spherically symmetric charge distributions, this law greatly simplifies the calculation of the electric field at any point. This concept is foundational for understanding and solving problems involving spherical capacitors, where the electric field varies with radius.

Comparison with Parallel-Plate Capacitors

Unlike parallel-plate capacitors, which typically exhibit a uniform electric field and energy density between the plates (ignoring edge effects), the energy density in a spherical capacitor is non-uniform due to the radial variation of the electric field. This concept highlights the importance of geometry in determining field distributions and energy storage characteristics in different capacitor configurations.

Transcript

00:01 In the given problem we have been given a spherical capacitor which comprises of two spherical shells as shown here one outer and one inner here the radii of these two shells are given as for inner one it is 12 .5 centimeter or we can see 12 .5 into 10 dash per minus 2 meter for the outer one here this is inner and this is outer for outer one this is 14 .8 centimeter or 14 .8 into 10 dash per minus 2 meter first of all we will find the capacitance of such a capacitor using the expression 4 pi epsilon not a b upon a minus b upon b minus a and when i put these values the capacitance of such a capacitor will come out to be 89 .4 pico ferrette now the potential across the plates across the shells has been given as 120 volt so now we will find charge stored over such a capacitor which is is given as c into v so it will be 89 .4 into 10 dashed bar minus 12 ferret into 120 volt so the charge is stored comes out to be 1 .074 into 10 dashed bar minus 8 pula now we have to find electric fields at two points first of the point is very close to the inertial at our distance r1 and second point is very close to the outer shell at a distance r2 and these distances are given as r1 is 12 .6 centimeter and r2 is 14 .7 centimeter we have to find energy densities at these two points so to find energy density first of all we will find electric field intensities at these two points to find electric field intensity we will use the expression 1 upon 4 pi epsilon not q by r square so for even it will become 9 into 10 ratio par 9 into q 1 .074 into 10 ratio to par minus 8 divided by 12 .6 into 10 ratio bar minus 2 to the whole square and this electric field comes out to be 6 ,0008 28 volt per meter.

03:13 Similarly we will find electric field at the second point 9 into 10 dash to the power 9 1 .074 into 10 dash to the power minus 8 and divided by 14 .7 into 10 dash to the power minus 2 the whole square and this electric field comes out to be 4 ,473 volt per meter so now as the first part of the problem, we can find energy density at first point, mu1, which comes out to be half epsilon not e square is its standard expression...

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