Question

The electric potential energy of a continuous charge distribution can be found in a way similar to that used for systems of point charges in Section \(23.6,\) by breaking the distribution up into suitable pieces. Find the electric potential energy of an arbitrary spherically symmetrical charge distribution, \(\rho(r) .\) Do not assume that \(\rho(r)\) represents a point charge, that it is constant, that it is piecewise-constant, or that it does or does not end at any finite radius, \(r\). Your expression must cover all possibilities. Your expression may include an integral or integrals that cannot be evaluated without knowing the specific form of \(\rho(r) .\) (Hint: A spherical pearl is built up of thin layers of nacre added one by one.)

Short answer

Based on the solution, the electric potential energy of a continuous spherically symmetrical charge distribution with density function ρ(r) can be expressed as: \(U = 2 \pi k_e \int_{0}^{R} r \rho(r) dr \int_{0}^{r} r'^2 \rho(r') dr'\). This expression is general and will need the specific form of ρ(r) and the total radius R of the distribution to be evaluated completely.

Step-by-step solution

Express the electric potential energy in terms of charge density function ρ(r)

First, consider a thin spherical shell of radius r and thickness dr within the spherically symmetrical charge distribution. The volume of this shell is given by: \(dV = 4 \pi r^2 dr\). The amount of charge within this shell can be expressed as: \(dq = \rho(r) dV\), where ρ(r) is the charge density function. The electric potential energy contained within this shell can be expressed as \(dU = \frac{1}{2} k_e dq \frac{dq'}{r}\), where \(k_e\) is the electrostatic constant and dq' is the charge enclosed by the shell with radius r. Now, we need to integrate this expression over the whole charge distribution to get the total electric potential energy.

Integrate the expression to get the total electric potential energy

To find the total electric potential energy of the entire distribution, we need to integrate the expression for dU over all the possible radii from 0 to R, where R is the total radius of the distribution: \(U = \int_{0}^{R} dU = \int_{0}^{R} \frac{1}{2} k_e \rho(r) dV \frac{dq'}{r}\). Now, substitute the expressions for \(dV\) and \(dq\) with respect to r from step 1: \(U = \int_{0}^{R} \frac{1}{2} k_e \rho(r) (4 \pi r^2 dr) \frac{\int_{0}^{r} \rho(r') 4 \pi r'^2 dr'}{r}\). We can simplify this expression by taking out constants and rewriting the integral: \(U = 2 \pi k_e \int_{0}^{R} r \rho(r) dr \int_{0}^{r} r'^2 \rho(r') dr'\). This is the expression for the electric potential energy of a spherically symmetrical charge distribution, ρ(r). This expression covers all possibilities and can only be evaluated if the specific form of ρ(r) is given.

Key concepts

Charge Distribution

Understanding charge distribution is crucial when analyzing electric potential energy. Charge distribution refers to how charges are arranged within a specific space. Charges can be distributed uniformly or non-uniformly, and their arrangement affects the electric fields and potential energies associated with them.

For example, in the case of a continuous charge distribution, we can't just consider each charge in isolation; instead, we use a charge density function, often denoted as \(\rho(r)\), to describe how the charge density varies with the distance from a reference point. This function \(\rho(r)\) provides all the information needed to calculate the electric properties of the charge distribution.

Considering that charges distribute differently in space, having the concept of charge density allows for a precise evaluation of the electric potential at any given point. For instance, when the charge density is high, the electric potential is usually stronger. Hence, understanding charge distribution helps us grasp the overall behavior of charges in a given environment.

Spherical Symmetry

The concept of spherical symmetry is fundamental in electrostatics, especially with respect to charge distributions. When a distribution has spherical symmetry, it means that the properties of the system are the same in every direction from the center. Importantly, this simplifies the mathematical treatment of the problem, as it allows us to focus on radial distances and disregard angular components.

If a charge distribution is spherically symmetrical, the charge density depends only on the distance from the center, not on the direction, or the angles \(\theta\) and \(\text{\phi}\). This implies that the charge density \(\rho(r)\) is a function of radius \(r\) alone. Such symmetry has profound implications for the resulting electric field, which will be purely radial and its magnitude will only depend on the distance from the center, too.

Employing spherical symmetry often results in more straightforward and general expressions in calculations, which hold true for any point system along the spherical shell. This symmetry greatly aids in the analytical process, making it a central concept when dealing with problems involving spherical charge distributions.

Charge Density Function

The charge density function, often represented as \(\rho(r)\), is a mathematical description used in electrostatics to model how charge is distributed in space. It explains the amount of charge per unit volume at any point in the charge distribution. For a spherically symmetric charge distribution, this function depends only on the radial distance from the center.

A high \(\rho\) value at a particular radius suggests a high concentration of charge at that distance from the center. Conversely, a low \(\rho\) may indicate that charge is sparse in that region. The function can take on various forms—it could be constant, vary linearly with \(r\), exponentially, or in another pattern entirely.

When calculating the electric potential energy in a system with a spherically symmetric charge distribution, one must integrate the charge density function over the volume of interest. The result of this integration, considering every infinitesimal volume of charge, provides a comprehensive view of the total charge within the domain. It’s a powerful tool that enables us to deal with continuous distributions instead of just discrete point charges.

Electrostatics

Electrostatics is the branch of physics that studies stationary or slow-moving electric charges. It encompasses the forces such charges exert on one another, the electric fields they generate, and the potential energy in these fields.

In the realm of electrostatics, the concept of electric potential energy is analogous to gravitational potential energy. Just as a mass in a gravitational field has potential energy depending on its height, a charge in an electric field has potential energy based on its position relative to other charges.

One fundamental principle of electrostatics is that like charges repel each other, while opposite charges attract. This principle means that moving a charge closer to a like charge requires work, increasing the system's electrical potential energy. Understanding electrostatics is essential for analyzing systems where charges are static or moving slowly enough that their motion doesn't create significant magnetic fields.

This field of physics is integral to problems involving electric potential energy and charge distributions, as these concepts are all related to how charges are stationary or slowly move in a system, affecting one another through electric fields and potentials.