Recent content by Screwdriver

Closer. The path difference is small when the detector is far away and then increases as d decreases. We'll "get to" m = 1 first. The intensity of a point source goes like 1/4 \pi r^2 , so if the intensity of the source is I_0, the intensity at the hole should be I_0/4 \pi x^2. Only a...

Sorry, I don't follow: m is just 1. Yes. Sure, but the intensity of the source is an overall constant which divides out. I assume the source is isotropic, so no. I guess not really, but my idea is that it's so small that it's close enough - otherwise I'm not sure what to do.

Homework Statement A point source of light (wavelength \lambda = 600 \, \text{nm} ) is located a distance x = 10\,\text{m} away from an opaque screen with a small circular hole of radius b. A very small photodiode is moved on an axis from very far away toward the screen. The first...

So what do those have to do with rotations in ##\mathbb{R}^3 ##?

One ##3##-dimensional representation of ##SO(3)## is ##3\times 3## rotation matrices parametrized by three Euler angles. The representation acts on ##3##-dimensional vectors and rotates them in ##\mathbb{R}^3 ##. That makes sense. What doesn't make sense is the interpretation of, say, the...

Problem This is a conceptual problem from my self-study. I'm trying to learn the basics of group theory but this business of representations is a problem. I want to know how to interpret representations of a group in different dimensions. Relevant Example Take SO(3) for example; it's the...

Okey dokey, thank you!

Homework Statement The picture of the circuit is attached; I want to find |V_{A}/V_{J}|. This seems really easy but I haven't done circuit analysis in forever. Homework Equations Complex impedances, Z_{C} = 1/i\omega C, Z_{R} = R. The Attempt at a Solution First R_{A} and C_{A} are in...

Ah yes, thank you! The dimensions were incorrect. Applying the correction, we find the same result. Also, no, I didn't account for the changing magnetic field as the cylinder speeds up; doing this with Faraday yields the same answer, so it's probably right. Thanks again to both of you :smile:

Homework Statement An infinite wire of linear charge density \lambda lies on the z axis. An insulating cylindrical shell of radius R is concentric with the wire and can rotate freely about the z axis. The charge per unit area on the cylinder is \sigma = -\lambda/2\pi R while the mass per unit...

Homework Statement $$ I=\int_{0}^{\infty} \frac{x^3}{e^x-1}\ln(e^x-1)\,dx $$ Homework Equations Any identity involving \zeta(s). The Attempt at a Solution I noted that: $$ \ln(e^x - 1) = \ln(x) + \sum_{n=0}^{\infty} \frac{B_{n+1}(1)}{(n+1)^2n!}x^{n+1} $$ Therefore, the integral...

There's a specific problem I'm doing, but this is more of a general question. The setup is a cylinder of mass m and radius R rolling without slipping down a wedge inclined at angle \alpha of mass M, where the wedge rests on a frictionless surface. I've made the Cartesian axis centred at the...

Ah, that does make a lot more sense. Thanks again everyone!

Hmm yes, that does seem to be of the same form. Honestly though, the material in the article seems way too advanced for the level that we're at (ie. the "do this because it works" stage) so I don't know if I'm even supposed to be worrying about this. You're probably right in that calling it the...

Thanks for the reply, I wasn't completely sure if this belonged in the advanced section or not. The system is an ##X_{6}## molecule with a single electron able to move between the different ##X## ions. The basis ket ##|e_{j}\rangle## represents the electron occupying the ##j^{\text{th}}## ion...