Problem 39 - Advanced Engineering Mathematics

Question

Coulomb's law states that the electric field \(\mathbf{E}\) due to a point charge \(q\) at the origin is given by \(\mathbf{E}=k q \mathbf{r} /\|\mathbf{r}\|^{3}\), where \(k\) is a constant and \(\mathbf{r}=x \mathbf{i}+y \mathbf{j}+z \mathbf{k}\). Determine the flux out of a sphere \(x^{2}+y^{2}+z^{2}=a^{2}\)

Step-by-Step Solution

Answer

The electric flux through the sphere is \(\frac{q}{\varepsilon_0}\).

Step 1: Recognize the Problem

We need to find the electric flux through a spherical surface due to a point charge located at the origin, using Coulomb's Law. The surface is defined by the sphere equation \(x^{2}+y^{2}+z^{2}=a^{2}\).

Step 2: Recall the Definition of Electric Flux

Electric flux through a closed surface is given by the formula \(\Phi = \oint \mathbf{E} \cdot d\mathbf{A}\), where \(d\mathbf{A}\) is the outward normal vector of the infinitesimal area element on the surface.

Step 3: Use Gauss's Law

Since the point charge is located at the center of the sphere, we can apply Gauss's Law directly. Gauss’s Law states that \(\Phi = \dfrac{q_{enc}}{\varepsilon_0}\), where \(q_{enc}\) is the total charge enclosed by the surface. Here, \(q_{enc} = q\).

Step 4: Calculate the Flux

Using Gauss's Law, the flux is \(\Phi = \dfrac{q}{\varepsilon_0}\). The symmetry of the problem and the spherical surface makes this possible without direct integration.

Step 5: Interpret the Result

The result shows that the flux is only dependent on the charge enclosed and not on the radius of the sphere or the position of the charge, as long as it is inside the surface.

Key Concepts

Electric Flux

Electric flux is a measure of the electric field passing through a given surface. It provides a sense of how much of the electric field is "flowing through" the surface in question. The concept of flux is often compared to the flow of a fluid through a net or a membrane: the more fluid that passes through, the greater the flux.

The electric flux through a closed surface is represented by the equation:

• \( \Phi = \oint \mathbf{E} \cdot d\mathbf{A} \)

where \( \mathbf{E} \) is the electric field and \( d\mathbf{A} \) is an infinitesimal area on the surface with an outward normal vector. The dot product \( \mathbf{E} \cdot d\mathbf{A} \) symbolizes how much of the field penetrates through the surface normally. A crucial aspect of this formula is integrating over the entire surface to accumulate the full effect of the electric field.

In cases like a spherical surface surrounding a point charge, this integral becomes more straightforward due to the system's symmetry. By determining the electric flux, you gain insight into how electric fields interact with surfaces, which is essential for understanding phenomena in electrostatics and electrodynamics.

Gauss's Law

Gauss's Law is a powerful tool in electrostatics, which relates the electric flux through a closed surface to the charge enclosed by that surface. It is elegantly expressed by the equation:

• \( \Phi = \dfrac{q_{enc}}{\varepsilon_0} \)

where \( \Phi \) is the electric flux, \( q_{enc} \) is the total charge enclosed within the surface, and \( \varepsilon_0 \) is the permittivity of free space.

This law significantly simplifies the calculation of electric fields when the problem exhibits symmetry. For instance, when a point charge is enclosed by a spherical surface, Gauss's Law allows you to find the flux without resorting to complex integration. Despite the surface's shape or size, if the charge remains enclosed, the flux depends solely on the charge, remarkable for its simplicity.

Gauss's Law helps visualize the field lines emanating from charges and is an essential principle in fields like electromagnetism and electrostatics. It allows for deeper insights into how charges and fields configure in different scenarios.

Point Charge

A point charge is an idealized model of a charged particle, treated as if it is a single point in space with a particular amount of charge. This model is useful for simplifying the analysis of electrostatic problems and is fundamental in understanding how charges interact.

According to Coulomb's Law, the electric field \( \mathbf{E} \) generated by a point charge \( q \) located at the origin is:

• \( \mathbf{E} = k\frac{q \mathbf{r}}{\|\mathbf{r}\|^{3}} \)

where \( k \) is a constant, and \( \mathbf{r} \) is a vector denoting the position relative to the charge.

The electric field radiates outward symmetrically from the point charge, decreasing with the square of the distance from the charge. This symmetry is particularly useful when applying Gauss's Law, as it allows simplifications otherwise difficult to achieve. Point charges are essential in theoretical studies and provide a foundation for understanding more complex charge distributions.

Spherical Surface

A spherical surface is a three-dimensional surface where every point is equidistant from a fixed central point - the sphere's center. When analyzing electrostatic problems, such as determining the electric flux through a surface, spherical surfaces offer advantages due to their inherent symmetry.

In the context of electrodynamics, a spherical surface surrounding a point charge provides a whole set of symmetry that simplifies computations. The radius of the sphere may change, but as long as the charge remains centered, the electric flux remains unaffected by the sphere's size. This highlights a neat property: the electric field behaves predictably within spherical symmetry.

Spherical surfaces are often used in problems involving point charges because they allow us to use Gauss's Law effectively. This makes solving problems more straightforward, helping students focus on understanding the electric field’s behavior rather than getting bogged down in the mathematics. Thus, understanding spherical surfaces is key to mastering electrostatics and other areas of physics.