Anaya Nolan
Gauss's law, formulated by Joseph-Louis Lagrange in 1773 and later by Carl Friedrich Gauss in 1835, is a problem-solving strategy that relates the surface integral of the normal component of an electric field to the charge enclosed within a Gaussian surface. It can be applied to deduce the electric field in cases where the charge occupies a finite volume, such as a spherical charge distribution. By constructing an imaginary closed surface (Gaussian surface) around a collection of charges, Gauss's law can be used to determine the electric field at that surface. This is particularly useful in highly symmetric situations and when dealing with conducting shells. The law is expressed mathematically using vector calculus in integral and differential forms, relating the electric field (E) to the total electric charge (Q) enclosed within a closed surface (S).
| Characteristics | Values |
|---|---|
| When Gauss's Law can be applied | When the charge occupies a finite volume, such as a spherical charge distribution |
| When there is symmetry in the problem, such as cylindrical, planar, or spherical symmetry | |
| When dealing with conducting shells | |
| When the charge distribution has spherical symmetry, i.e., the density of charge depends only on the distance from a point in space and not on the direction | |
| When the electric field is either parallel or perpendicular to the surface vector | |
| When the electric field passes through the surface in a uniform way | |
| When the total flux is known | |
| When the charge is inside a Gaussian surface | |
| When the electric field at a point outside or inside a shell needs to be found | |
| When the charge enclosed depends on the distance of the field point relative to the radius of the charge distribution | |
| When the charge is on the axis of a disk, and then taking the limit as the radius of the disk approaches infinity | |
| When the cross-sectional area of the cylinder is arbitrarily small |
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What You'll Learn
Gauss's Law and electric fields
Gauss's law relates the surface integral of the normal component of an electric field to the charge enclosed inside. It can be used to calculate the electric field in cases where the charge occupies a finite volume.
Gauss's law can be applied to deduce the electric field in cases where there is symmetry. There are three types of symmetry that allow Gauss's law to be used: cylindrical symmetry, planar symmetry, and spherical symmetry. In the case of spherical symmetry, the density of charge depends only on the distance from a point in space and not on the direction. For example, if a sphere of radius R is uniformly charged, it has spherical symmetry.
The direction of the field at a point P depends on whether the charge in the sphere is positive or negative. When the charge in the sphere is positive, the direction is from O to P. On the other hand, if the charge is negative, the direction is from P to O.
Gauss's law can be used to calculate the electric field in a volume bounded by a closed surface when there is a net charge inside. If there is no net charge inside the closed surface, then the electric flux through the surface is zero.
The law can be expressed mathematically using vector calculus in integral form and differential form, relating the electric field E and the total electric charge, or the electric displacement field D and the free electric charge.
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Net electric flux
Gauss's Law relates the surface integral of the normal component of an electric field to the charge enclosed inside a Gaussian surface. The law can be applied to deduce the electric field in three types of symmetry: spherical symmetry, cylindrical symmetry, and planar symmetry.
The electric flux through a surface is proportional to the number of field lines crossing that surface. The numerical value of the electric flux depends on the magnitudes of the electric field and the area, as well as the relative orientation of the area with respect to the direction of the electric field.
The direction of the field at point P depends on whether the charge in the sphere is positive or negative. For a net positive charge enclosed within the Gaussian surface, the direction is from O to P, and for a net negative charge, the direction is from P to O.
The flux of an electric field through a shaded area captures information about the "number" of electric field lines passing through the area. The amount of flow through a hoop in a flowing river depends on the angle of the hoop relative to the direction of the current, the strength of the current, and the size of the hoop.
The net flux through a cube placed between two charged plates is zero. The electric flux through the bottom face is negative, and through the top face is positive, while the electric flux through the other faces is zero since the electric field is perpendicular to the normal vectors of those faces.
The electric flux through the area of the i-th patch can be calculated using the formula:
\[\co: 11>\Phi_i = \vec{E}_i \cdot \delta \vec{A}_i \ \ (i \mathrm{th \, patch}).
\]
The flux through each individual patch can be added to give an estimate of the net flux through the entire surface.
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Charge distribution
Gauss's law can be used to find the distribution of electric charge. It is one of Maxwell's equations and forms the basis of classical electrodynamics. The law can be expressed mathematically using vector calculus in integral and differential forms, both of which are equivalent due to the divergence theorem, also known as Gauss's theorem.
The electric flux through any hypothetical closed surface is equal to the net electric charge enclosed within that surface multiplied by 1/ε0, where ε0 is the electric constant. This closed surface is referred to as a Gaussian surface. Gauss's law can be applied to find the electric field produced by a charge distribution, but only if there is enough symmetry in the charge distribution.
There are three types of symmetry that allow Gauss's law to be used to deduce the electric field: spherical, cylindrical, and planar symmetry. For example, in the case of an infinite line and cylinder, a cylinder with end-caps perpendicular to the line or cylinder is used as the Gaussian surface. For an infinite slab, a cylinder with its shaft perpendicular to the slab is used.
The charge distribution in atoms is assumed to be spherical, allowing us to discuss s, p, and d orbitals. This spherical symmetry results in the field being directed radially outwards and remaining constant on any sphere sharing the same centre as the charge distribution.
Gauss's law can also be applied to chemical systems, such as a charged bilayer or a plane electrode. In these cases, the charge is spread uniformly on an infinite plane with a charge density of σ per unit area, resulting in the field being directed away from the plane in the ±z directions.
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Cylindrical symmetry
Gauss's law is a powerful tool for determining expressions for the electric field, even though it is not directly about the electric field but rather electric flux. In scenarios with specific symmetries (spherical, cylindrical, or planar) in the charge distribution, Gauss's law can be applied to deduce the electric field based on knowledge of the electric flux.
For instance, a uniform charge density in an infinitely long straight wire or cylinder demonstrates cylindrical symmetry. Conversely, an infinitely long cylinder with varying charge densities along its length or with charge densities that change with direction does not possess cylindrical symmetry.
When dealing with cylindrical symmetry, a cylindrical Gaussian surface is typically employed. This approach simplifies Gauss's law, making it easier to calculate the flux through the surface. The electric field in a cylindrically symmetrical scenario is solely dependent on the distance from the axis, with the direction of the field pointing away from the axis for positive charges and towards it for negative charges.
To apply Gauss's law effectively, it is essential to select a suitable Gaussian surface that aligns with the symmetry of the problem. This strategic choice simplifies the calculation of the flux integral. For example, when addressing a line of charge, a cylindrical Gaussian surface is a suitable option. By exploiting the symmetry of the system, Gauss's law becomes a valuable tool for determining the electric field in various scenarios with high symmetry.
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Curved surfaces
Gauss's Law can be applied to curved surfaces, such as cylinders and spheres, to determine the electric field or flux produced by a charge distribution. This is done by selecting a Gaussian surface that encloses the charge and applying the law to calculate the electric flux passing through the surface.
In the case of a cylinder, the Gaussian surface consists of a disk at each end and the side of the cylinder. When applying Gauss's Law, there are three contributions to the flux: one from each end and one from the curved surface. However, if the electric field (E) and the differential vector area (dA) are perpendicular, their dot product becomes zero, resulting in no contribution to the flux. This is often the case for the ends of the cylinder, where E and dA are perpendicular, while on the curved surface, they are typically parallel, contributing to the flux.
For a sphere, Gauss's Law can be applied by considering a small area on its surface. By rotating this area to different angles, the number of field lines passing through it changes, but the corresponding "area" seen by the field lines also adjusts proportionally. This relationship allows for the conservation of flux density, regardless of the shape of the surface. Therefore, the sphere can be mentally manipulated into different shapes while still adhering to Gauss's Law.
The choice of the Gaussian surface depends on the symmetry of the problem. Common symmetries that lend themselves to Gauss's Law include cylindrical symmetry, planar symmetry, and spherical symmetry. By exploiting these symmetries, the direction of the electric field (E) can be determined, and a suitable Gaussian surface can be chosen to simplify the application of the law.
In summary, Gauss's Law can be applied to curved surfaces by selecting a Gaussian surface that encloses the charge distribution and considering the symmetry of the problem. The electric flux passing through the surface can then be calculated, taking into account the contributions from different components of the surface, such as the ends and curved surface of a cylinder or the adjustable shape of a sphere.
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Frequently asked questions
Gauss's law relates the surface integral of the normal component of an electric field to the charge enclosed within it. It was formulated by Joseph-Louis Lagrange in 1773 and later by Carl Friedrich Gauss in 1835.
Gauss's law can be applied when there is symmetry in the problem, allowing the electric field to pass through the surface uniformly. Common examples include cylindrical symmetry, planar symmetry, and spherical symmetry.
Gauss's law has several practical applications, including computing electric fields in highly symmetric situations and dealing with conducting shells. It can also be used to find the distribution of electric charge in a conductor.
