# Use Gauss' Law to find the electric field

• latentcorpse
In summary, the conversation discusses using Gauss' Law to find the electric field and potential of a charge with uniform density in a specific region. The speaker also mentions integrating over different spheres and using the concept of symmetry to solve the problem.
latentcorpse
Use Gauss' Law to find the electric field, everywhere,of charge of uniform density $\rho$ occupying the region $a<r<b$, where r is the distance from the origin. Having done this, find the potential.

Ok, so far I said that by Gauss' Law,

$\Phi=\oint_S \vec{E} \cdot \vec{dS} = \int_V \nabla \cdot \vec{E} dV = \frac{Q}{\epsilon_0} = \frac{1}{\epsilon_0} \int_V \rho dV$

and since V is arbitrary I obtain Poisson's Equation
$\nabla \cdot \vec{E} = \frac{\rho}{\epsilon_0}$

I just can't see how to rearrange for E?

Initially I was considering integrating something over a sphere of radius b and then over a sphere of radius a and subtracting them but I don't have any idea what to integrate.

Are either of these ideas useful? If so, what do I do next? If not, can you suggest something?

Cheers

What is a and b? If you are calculating the E field inside the uniformly charged sphere then the charge enclosed by the Gaussian surface is a function of the radius. Outside the sphere it can be treated as a point charge. The potential then can be found between the points a and b.

Don't ignore the most important point of the problem. There is a symmetry that tells you the direction of E (+ or -) at every point in space, and that the magnitude of E is independent of two certain generalized coordinates.

## 1. What is Gauss' Law?

Gauss' Law is a fundamental principle in electromagnetism that relates the electric field at a point to the charge enclosed by a surface surrounding that point. It was first formulated by German mathematician and physicist Carl Friedrich Gauss in the early 19th century.

## 2. How do I use Gauss' Law to find the electric field?

To use Gauss' Law, you first need to identify a closed surface that encloses the point where you want to find the electric field. Then, you need to calculate the electric flux through that surface, which is the product of the electric field and the surface area. Finally, you can use the equation E∫dA = Qenc/ε0, where E is the electric field, dA is the surface area, Qenc is the enclosed charge, and ε0 is the permittivity of free space, to solve for the electric field.

## 3. What is the significance of Gauss' Law?

Gauss' Law is significant because it is one of the four Maxwell's equations that form the foundation of classical electromagnetism. It allows us to calculate the electric field at a point without needing to consider the individual charges that contribute to that field. It also helps us understand the symmetry of electric fields and how they are affected by the distribution of charges.

## 4. Can Gauss' Law be applied to any shape of surface?

Yes, Gauss' Law can be applied to any closed surface, regardless of its shape. However, it is important to choose a surface that has a high degree of symmetry to simplify the calculation of the electric flux. For example, a spherical surface is often used because it has uniform symmetry in all directions.

## 5. Are there any limitations to using Gauss' Law?

While Gauss' Law is a powerful tool for calculating electric fields, it does have some limitations. It only applies to static electric fields, meaning that the charges must be stationary and not changing over time. It also assumes that the materials involved are linear and isotropic, meaning that the electric field is directly proportional to the charge and is the same in all directions.

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