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Mobile App Chapter 23: Problem 67
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 the sphere that is contained within the sphere is \( (\frac{r}{R})^5 \).
Step by step solution
01
Define given
The sphere contains a volume charge density \( \rho \). The total electrostatic energy of the sphere is then given by \( U = \frac{1}{2}\int\rho V d\tau \), where \( V \) is the electric potential and \( d\tau \) is the volume element for integration.
02
Calculate Electric Potential
The aim is to find the electric potential \( V \). From Gauss's law (for a solid sphere), we know that the electric field \( E \) within the sphere is given by \( E = \frac{\rho r}{3\epsilon_0} \), where \( r \) is the distance from the centre of the sphere and \( \epsilon_0 \) is the permittivity of free space. The electric potential \( V \) at distance \( r \) is the negative integral of the electric field from infinity to \( r \), i.e., \( V = -\int_{\infty}^{r} E dr = -\int_{\infty}^{r} \frac{\rho r}{3\epsilon_0} dr = \frac{\rho r^2}{6\epsilon_0} \). This is the electric potential for points inside the sphere.
03
Calculate Energy Density
The energy density i.e., energy per unit volume is given by \( \frac{1}{2}\rho V \), therefore substituting \( V \) from above, we get: \( u = \frac{1}{2} \rho \cdot \frac{\rho r^2}{6\epsilon_0} = \frac{\rho^2 r^2}{12\epsilon_0}\).
04
Calculate Total Electrostatic Energy
The total electrostatic energy \( U \) contained within the sphere is the integration of energy density over the entire volume of the sphere: \( U = \int_{0}^{R} u d\tau = \int_{0}^{R} \frac{\rho^2 r^2}{12\epsilon_0} 4\pi r^2 dr = \frac{\rho^2}{12\epsilon_0} 4\pi \int_{0}^{R} r^4 dr = \frac{\rho^2}{12\epsilon_0} 4\pi \frac{R^5}{5} = \frac{\rho^2 R^5 \pi}{15\epsilon_0} \). This is the total electrostatic energy of the sphere.
05
Calculate Fraction of Energy within Sphere
As specified in the question, 'within the sphere' means a distance \( r \) from the centre. The total energy within a radius \( r \) is then: \( U_r = \frac{\rho^2 r^5 \pi}{15\epsilon_0} \). The asked fraction is then \( \frac{U_r}{U} = \frac{\frac{\rho^2 r^5 \pi}{15\epsilon_0}}{\frac{\rho^2 R^5 \pi}{15\epsilon_0}} = (\frac{r}{R})^5 \).
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Volume Charge Density
Volume charge density, denoted as \( \rho \), is a measure of how much electric charge is distributed throughout a given volume of space. It is defined as the amount of charge \( Q \) per unit volume \( V \), which is mathematically expressed as \( \rho = \frac{Q}{V} \
\). In the context of a uniformly charged solid sphere, the volume charge density is constant throughout the object, making the calculations of the electric field and potential simpler. This concept is critical for understanding how charge is distributed in three-dimensional objects and is a prerequisite for calculating the electrostatic energy stored within such objects.Understanding the value of \( \rho \) is also essential when applying Gauss's law to find the electric field inside the sphere. Moreover, knowing the volume charge density allows us to determine the energy density and consequently the total electrostatic energy of the sphere using integration techniques.
Thus, volume charge density is not just a theoretical concept, but a practical tool in the analysis and computation of electrostatic phenomena in charged objects like spheres.
\). In the context of a uniformly charged solid sphere, the volume charge density is constant throughout the object, making the calculations of the electric field and potential simpler. This concept is critical for understanding how charge is distributed in three-dimensional objects and is a prerequisite for calculating the electrostatic energy stored within such objects.Understanding the value of \( \rho \) is also essential when applying Gauss's law to find the electric field inside the sphere. Moreover, knowing the volume charge density allows us to determine the energy density and consequently the total electrostatic energy of the sphere using integration techniques.
Thus, volume charge density is not just a theoretical concept, but a practical tool in the analysis and computation of electrostatic phenomena in charged objects like spheres.
Gauss's Law
Gauss's law is one of the four Maxwell's equations, formulating the relationship between electric charge and the resulting electric field. The law states that the net electric flux through a closed surface is equal to the charge enclosed by the surface divided by the permittivity of the medium. Mathematically, Gauss's law is expressed as \( \Phi_E = \frac{Q_{\text{enc}}}{\varepsilon_0} \), where \( \Phi_E \) is electric flux, \( Q_{\text{enc}} \) is the enclosed charge, and \( \varepsilon_0 \) is the permittivity of free space.
When considering a charged sphere with a uniform volume charge density, Gauss's law is used to calculate the electric field at any point within the sphere. The symmetry of a spherical charge distribution simplifies the application of Gauss's law, allowing us to determine that the electric field at a distance \( r \) from the center of the sphere is proportional to \( r \) and the charge density \( \rho \). The ease of applying Gauss's law in spherical geometries makes it an invaluable tool in our electrostatic toolkit.
When considering a charged sphere with a uniform volume charge density, Gauss's law is used to calculate the electric field at any point within the sphere. The symmetry of a spherical charge distribution simplifies the application of Gauss's law, allowing us to determine that the electric field at a distance \( r \) from the center of the sphere is proportional to \( r \) and the charge density \( \rho \). The ease of applying Gauss's law in spherical geometries makes it an invaluable tool in our electrostatic toolkit.
Electric Potential
Electric potential, often simply called potential, is the amount of electric potential energy per unit charge at a specific point in the field. The potential at a distance \( r \) from a charge distribution is determined by the work done in bringing a unit positive charge from infinity to that point against the electric field.For the charged sphere with uniform volume charge density, we calculate potential by integrating the electric field from infinity to the point of interest. This process yields a formula for the potential inside the sphere, which, alongside the volume charge density, is crucial for determining the energy density and, by extension, the total electrostatic energy of the sphere.
Electric potential is a scalar quantity, unlike the electric field, which is a vector. This distinction is beneficial in the context of our problem, as the scalar nature of electric potential eliminates the need to consider direction, simplifying the calculation of the electrostatic energy contained within the charged sphere.
Electric potential is a scalar quantity, unlike the electric field, which is a vector. This distinction is beneficial in the context of our problem, as the scalar nature of electric potential eliminates the need to consider direction, simplifying the calculation of the electrostatic energy contained within the charged sphere.
