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Chapter 23: Problem 63

A solid sphere contains a uniform volume charge density. What fraction of the total electrostatic energy of this configuration is contained within the sphere?

Short Answer

Expert verified
The fraction of the total electrostatic energy of this configuration contained within the sphere can be calculated by dividing the integral of the energy within a sphere of radius r inside the given sphere by the total energy. The specific result depends on values for total volume and charge density which are not provided in the exercise.

Step by step solution

01

Identifying Given Values

Identify the volume V of the sphere and the charge density \( \rho \) which is uniformly distributed. For this exercise, no direct numerical values are given, but we can proceed using these symbols.
02

Total Charge Calculation

Calculate the total charge Q of the sphere using the formula for the volume of a sphere \( V = \frac{4}{3} \pi r^3 \) and the definition of volume charge density \( Q=\rho V \). So, the total charge \( Q = \rho \cdot \frac{4}{3} \pi r^3 \)
03

Total Electrostatic Energy Calculation

The total electrostatic energy W of the sphere can be considered as the energy of charging up the sphere from 0 to Q. This is given by the formula \( W = \frac{1}{2} Q^2/R \) where R is the distance from the center to the surface of the sphere.
04

Energy Within the Sphere Calculation

Calculate the electrostatic energy within a sphere of radius r which is inside the given sphere. Consider a volume element dv inside the sphere at a distance r from the center, and integrate from 0 to R, given by \( \frac{1}{2} \int_{0}^{R} \frac{Q}{R}^2 dv \). Here dv represents a small volume of the sphere.
05

Fraction of Total Energy Calculation

Finally, calculate the fraction of the total energy contained within the sphere by dividing the Energy within the sphere from step 4 by the Total Electrostatic Energy from step 3.

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Most popular questions from this chapter

Two positive point charges are infinitely far apart. Is it possible, using a finite amount of work, to move them until they're a small distance \(d\) apart?

Two capacitors are connected in series and the combination is charged to 100 V. If the voltage across each capacitor is \(50 \mathrm{V}\) how do their capacitances compare?

Does the capacitance describe the maximum amount of charge a capacitor can hold, in the same way that a bucket's capacity describes the maximum amount of water it can hold? Explain.

Engineers testing an ultracapacitor (see Application on page 420 ) measure the capacitor's stored energy at different voltages. The table below gives the results. Determine a quantity that, when you plot stored energy against it, should give a straight line. Make your plot, establish a best-fit line, and use its slope to determine the capacitance. $$\begin{array}{|l|l|l|l|l|l|l|l|} \hline \text { Voltage (V) } & 12.2 & 20.1 & 31.8 & 37.9 & 45.7 & 50.2 & 56.0 \\\ \hline \text { Energy (kJ) } & 9.25 & 27.2 & 62.5 & 94 & 139 & 158 & 203 \\ \hline \end{array}$$

Capacitors \(C_{1}\) and \(C_{2}\) are in series, with voltage \(V\) across the combination. Show that the voltages across the individual capacitors are \(V_{1}=C_{2} V /\left(C_{1}+C_{2}\right)\) and \(V_{2}=C_{1} V /\left(C_{1}+C_{2}\right)\)
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