In advanced physics the concept of symmetry plays a crucial...

In advanced physics, the concept of symmetry plays a crucial role in understanding fundamental interactions and the laws governing the universe. Can you explain how Noether's theorem connects symmetries and conservation laws in theoretical physics? Specifically, how does the presence of continuous symmetries in a physical system lead to conserved quantities, and what are some practical examples of this connection in classical mechanics, quantum mechanics, and field theory? Additionally, how does gauge symmetry extend this concept in the context of the Standard Model of particle physics, contributing to our understanding of fundamental forces?

Transcript

00:01 In this problem, we're going to talk about galsey's law.

00:04 So first, before actually diving into the problem, i'm going to give you a brief review of the law.

00:11 So suppose that we have a charge density here, and that the total charge contained here is equal to q.

00:22 So here we have a charge, here we have q.

00:26 And i suppose that our goal is to find what is the value of the electric field.

00:32 At a distance, let's say, r from the center of mass of this charge.

00:40 And let's say that we can, we draw a surface around this charge.

00:48 This is a 3d surface, okay? actually, the surface itself is 2d, but it's embedded in the 3d space.

00:56 Here i'm only drawing 2d because of my drawing limitations.

01:00 And then what gauss's law tells us is that if you take the surface integral of the electric field over this gaussian surface around the charge density, the integral of the electric field over the surface s, then this is equal to epsilon zero, one over epsilon zero, i'm sorry, one over the the vacuum permittivity times the charge density.

01:42 This is the integral form of galses law.

01:49 There's also the differential form.

01:54 We have the divergent of e.

02:00 This is equal to 1 over epsilon 0 times the density of charges.

02:08 And in our problem, what we have to do is to discuss what is the role that symmetry place in gossus law.

02:16 Why it's so important for us that the charge distributions we're working with be symmetric.

02:26 Well, notice that what we're really solving is this equation here, this integral equation, and it's not always easy to solve this equation.

02:39 So notice that we want to find e, but in order to solve the equation, we have to integrate e, even though we don't know what it is.

02:50 So there must be some constraint that we have to input to our system, such that we can somehow isolate e and find it.

03:01 In the case, when we're working with symmetric distributions, let's say that we have a symmetric charge density.

03:11 So let's say that the charge density is located in a sphere, as you're in my drawing, let's say this is a sphere that has a total charge that has a total charge q...

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