Kara Sears
Gauss's law, also known as Gauss's flux theorem, is one of Maxwell's equations that relate the distribution of electric charge to the resulting electric field. It can be used to determine the electric field across a surface enclosing any charge distribution if there is symmetry in the problem. However, it cannot be used to find the electric field of a finite-length charged wire because it lacks the symmetry required to obtain an exact solution. This is because the field does not have a constant magnitude or direction over the line, resulting in 'edge effects' where the E-field bends towards the endpoints of the wire. Nonetheless, Gauss's law can be applied to infinite charge distributions, where it is very useful as the field is uniform and there are no edge effects.
| Characteristics | Values |
|---|---|
| Applicability | Gauss's law applies to all electrostatic cases. |
| Usefulness | Gauss's law is not useful for finite wires/lines of charge as it cannot find the flux without calculating an integral. |
| Exceptions | If the line is very long, Gauss's law can be used to find the field near the middle of a charged line. |
| Integral Form | The flux of the electric field out of a closed surface is proportional to the electric charge enclosed by the surface. |
| Differential Form | The divergence of the electric field is proportional to the local density of charge. |
| Symmetry | Common examples of symmetries that lend themselves to Gauss's law include cylindrical symmetry, planar symmetry, and spherical symmetry. |
| Relation to Coulomb's Law | Gauss's law contains Coulomb's law, so using Coulomb's law is essentially the same as using Gauss's law. |
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What You'll Learn
Gauss's Law for finite linear conductors
Gauss's law, also known as Gauss's flux theorem or Gauss's theorem, relates the distribution of electric charge to the resulting electric field. It was first formulated by Joseph-Louis Lagrange in 1773 and later by Carl Friedrich Gauss in 1835 in the context of the attraction of ellipsoids.
Gauss's law can be applied to charge distributions that are finite. However, the key to applying Gauss's law is that the field must be uniform over the Gaussian surface, meaning it must have the same magnitude and direction on the surface. In the case of a finite line, the field does not have a constant magnitude or direction, and hence, the field over a Gaussian cylinder would not be constant. Thus, it is not possible to use Gauss's law to find the electric field of a finite-length charged wire.
However, if the line is very long, and you are interested in the field near the middle of the charged line, then the field would be approximately constant over a Gaussian cylinder of finite length, and you can use Gauss's law to obtain the field near the center. Additionally, in cases where symmetry mandates uniformity of the field, Gauss's law can be used in its differential form, which states that the divergence of the electric field is proportional to the local density of charge.
Furthermore, Coulomb's law can be derived using Gauss's law, and the charge in any given region of the conductor can be deduced by integrating the electric field to find the flux through a small box perpendicular to the conductor's surface. The total flux of the electric field through a closed surface outside the conductor is zero, and hence, the total charge inside the closed surface is also zero.
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Electric field symmetry
Gauss's Law, also known as Gauss's flux theorem, is one of Maxwell's equations and relates the distribution of electric charge to the resulting electric field. It states that the flux of the electric field out of a closed surface is proportional to the electric charge enclosed by the surface, irrespective of the charge distribution.
While the law is insufficient to determine the electric field across a surface enclosing any charge distribution, it can be used when symmetry mandates uniformity of the field. Common examples of symmetries that lend themselves to Gauss's law include cylindrical symmetry, planar symmetry, and spherical symmetry.
In the case of a finite wire, Gauss's law is not useful as the translational symmetry along the wire is lost, and the form of the electric field is unpredictable. However, for an infinite wire, symmetry assumptions can be made, and Gauss's law becomes applicable.
Planar symmetry, for instance, occurs when charges are uniformly spread over a large flat surface. In this case, the electric field at a given point can only depend on the distance from the plane and has a direction either towards or away from it. The electric flux through a curved surface is zero, and the flux through the flat ends equals the electric field magnitude multiplied by the surface area. By combining these facts with Gauss's law, the electric field magnitude can be determined.
Therefore, while Gauss's law is not always applicable for finite elements, it can be used when the problem exhibits certain symmetries that allow for the assumption of a uniform electric field.
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Cylindrical symmetry
Gauss's Law is a powerful tool for determining expressions for the electric field, even though it is primarily about electric flux. In cases where there is spatial symmetry in the charge distribution (spherical, cylindrical, or planar), Gauss's Law can be used to find the electric field by examining the electric flux.
However, if the charge density varies along the length of the cylinder or changes with direction, it loses its cylindrical symmetry. For example, if a cylinder has different charge densities at different positions along its length, it does not possess usable cylindrical symmetry. Similarly, if the charge density varies with direction, such as having different charge densities for different angular positions, cylindrical symmetry is not applicable.
To apply Gauss's Law in cases of cylindrical symmetry, a closed Gaussian surface in the shape of a cylinder is chosen. This Gaussian cylinder has the same axis as the axis of the charge distribution. The electric field at any point on this surface displays cylindrical symmetry and is directed radially outwards for positive charges and inwards for negative charges. The magnitude of the electric field remains constant on the surface of the cylinder.
The flux through the Gaussian cylinder can be calculated by considering the flux through the flat ends and the curved surface separately. The flux through the end caps is zero because the area vector is perpendicular to the electric field. By applying Gauss's Law, the flux is equal to the charge enclosed within the volume divided by the permittivity of free space. This allows us to determine the magnitude of the electric field at a distance from the axis of the cylinder.
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Planar symmetry
Gauss's law is a powerful tool for determining expressions for the electric field, even though it primarily concerns electric flux. In scenarios with specific symmetries, such as spherical, cylindrical, or planar symmetries, we can utilise Gauss's law to ascertain the electric field by comprehending the electric flux. These symmetries enable us to identify a Gaussian surface where the electric field maintains a consistent magnitude.
Now, let's delve into planar symmetry. In the context of planar symmetry, a flat surface uniformly distributes charges, resulting in identical charge characteristics at all points within a plane parallel to the charged plane. This implies that the electric field at a given point is solely influenced by the distance from the plane, either towards or away from it. Consequently, the electric field at this point exhibits a non-zero z-component.
To calculate the electric field due to a planar charge distribution with surface charge density σ, we can employ Gauss's Law. In this scenario, a suitable Gaussian surface is a box that intersects the plane, featuring two faces parallel to the plane and the remaining sides perpendicular. The flux through the parallel sides of the box is zero since the normal vector to these sides aligns with the plane, contrasting with the electric field, rendering the product of their dot product zero.
The electric field is perpendicular to the top and bottom of the box, maintaining a constant magnitude. By integrating the electric field over the top or bottom surface, we can determine the flux through one of these faces. Due to symmetry, we recognise that the flux through the top and bottom of the box is identical. This understanding aids in calculating the charge enclosed by the box, which is the product of the surface charge density σ and the area of the plane inside the box.
By substituting the obtained values into Gauss's Law, we can establish a relationship between the electric field, the charge enclosed, and the permittivity of free space. This allows us to determine the magnitude of the electric field. It's important to acknowledge that the assumption of an infinite plane leads to the independence of the electric field from the distance to the plane. However, for finite planes near the centre, this assumption still offers a valuable approximation.
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Spherical symmetry
Gauss's Law is a powerful tool for determining the electric field in situations with specific symmetries, such as spherical, cylindrical, or planar charge distributions. It is not the shape of the object but the charge distribution that determines spherical symmetry.
A charge distribution exhibits spherical symmetry if the density of the charge depends solely on the distance from a point in space and not on the direction. In other words, if you rotate the system, it should appear unchanged. For instance, a sphere with a uniform charge density possesses spherical symmetry. On the other hand, if a sphere has distinct charge densities in its top and bottom halves, it lacks spherical symmetry because the charge density depends on the direction.
In the context of Gauss's Law, spherical symmetry implies that the electric field has only a radial direction component. This means that the electric field vectors are directed radially outwards, normal to the surface. This symmetry allows for the simplification of calculations by choosing an appropriate Gaussian surface that matches the symmetry of the charge distribution. By doing so, the integral over the Gaussian surface can be evaluated, and the flux through the surface can be determined.
Gauss's Law is particularly useful when dealing with finite volumes of charge distribution. For example, when considering a spherical charge distribution, it is interesting to explore what the electric field is like inside the distribution. By applying Gauss's Law, we can determine the electric field at a point P that is a distance r from the centre of the distribution. The magnitude and direction of the electric field at this point can be calculated, and the direction depends on whether the charge within the sphere is positive or negative.
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Frequently asked questions
No, Gauss's Law cannot be used to find the Electric Field of a finite-length charged wire. This is because there is not enough symmetry to get an exact solution.
Symmetry is required because the field must be uniform over the Gaussian surface, with the same magnitude and direction on the surface.
Yes, if the line is very long, you can use Gauss's Law to obtain the field near the centre.
Gauss's Law can be used to find the Electric Field of an infinite-length charged wire.
Yes, Coulomb's Law can be used to solve for the Electric Field of a finite-length charged wire.
