Review problem. Consider a model of the nucleus in which the positive charge (Z e) is uniformly

Review problem. Consider a model of the nucleus in which the...

Key Concepts

Electric Field Energy Density

This concept refers to the amount of energy stored per unit volume in an electric field, expressed as (1/2) ??E². It forms the basis to calculate the energy stored in a field by integrating over a region of space, and is a crucial concept in both classical electromagnetism and energy analysis within charged systems.

Uniform Charge Distribution

This concept involves a charge that is spread out evenly over a volume, such as a sphere. Understanding the uniform distribution is essential, as it simplifies the problem by providing symmetry, which in turn permits the derivation of the electric field and potential inside and outside the distribution using Gauss's Law.

Electric Field in a Uniformly Charged Sphere

Within a uniformly charged sphere, the electric field behaves differently inside and outside the sphere. Inside, the field increases linearly with distance from the center, while outside, it behaves as if all the charge were concentrated at the center. This distinction is pivotal when integrating energy densities over space to obtain the total energy stored in the configuration.

Integration Over All Space

To compute the total electrostatic potential energy stored in the field, one must integrate the energy density over all space. This involves setting up integrals in spherical coordinates that account for regions with different functional forms of the electric field, thereby allowing a complete evaluation of the energy associated with a charge distribution.

Transcript

00:01 In this question, we want to actually show that the electrostatic energy for a nuclei or nucleus is given as shown in the question.

00:20 Then we want to prove that or show that equation by integrating the energy density over all space.

00:27 So the energy density is defined as half epsilon knot times e square, where e is the electric field at that particular point.

00:45 Now we're going to assume that our nucleus is a perfect sphere, radii of r, and the integral that we will need is cross half epsilon knot e square overall space which we can either choose to integrate dx, d -x, d -y from x to 0 to infinity as well as y from 0 to infinity and also dz because it's a three -dimensional space but of course this is too complicated we can also do the integral in radium coordinate so this will be 0 to infinity and radial coordinates we'll be integrating 4 pi r square the r this is our our small units of interval for integration basically kind of like adding layers of spheres right with 3di of r small r small r over from r equals to zero to r equals to infinity so this is what we are doing we want to find this integral but to do that we need to know what is our e right the electric field to find the electric field we need to use gals law right where gaucious law states that the total electric flux passing through a surface is proportional to the amount of charge that is contained within that surface.

02:53 So the surface must be a closed surface, of course.

02:57 In this case, we're going to choose a closed surface that is a sphere.

03:02 This is because of symmetry reasons, right? so the entire flux coming out from a surface, say, at a distance of r, called its distance r.

03:19 Total flux will be the electric field, multiplied by 4 pi r square, the surface area.

03:28 This is proportional, with the proportionality constant 1 over epsilon to the total charge that is contained inside.

03:41 Now for the total charge that's contained, this would be the same.

03:52 Throughout when we are considering from r equals to the radius, big r, all the way to infinity.

04:02 It will be the same total charge that is encompassed within that sphere.

04:08 However, that would not be the same when we are considering from 0 to big r, right, from 0 to big r.

04:20 This will, the amount of charge that it actually encompassed would vary as you move more and more outwards, right? this will depend on the density of the charge that is in the nucleus.

04:40 So if you had to consider for r less than the radius speak r, the amount of charge that is encompassed within it, we will need to take the density, the charge density, of our nucleus, multiply by 4 .3 pi r cubed.

05:03 What is the density? the charge density will be taking the total charge, that is set times e, divided by the total volume, which is 4 .3 pi r cubed.

05:20 So this is our charge density.

05:26 Simplifying this we should get ze times r q over big r cube...

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