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I understand that light passed through the objective lens of the telescope produce an image that is real and inverted, and have a height which is measured as the distance between the focal point and the principal axis, but my question is: Why? If all the light is concentrated onto the focal point, then why does the height matter, because wouldn't the light rays coming from a given object be focused on to a single point? Please help me understand the lapses in my logic.

Thanks for any help you can give me, I really appreciate it!

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if we have two charged insulators. One positively charged and another negatively. if we connect them together, their charges transfer from one to another or not?

transfer charge with conducting occur or not?

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I am trying to calculate some berry curvature (BC) in a 2D lattice and I have some things I am getting lost with.

In the 2D lattice, we set up the eigenvalue problem $H|u_1\rangle = \epsilon_i|u_i\rangle$. Numerically, I can find $\epsilon,u$ by diagonalizing some matrix. This gives my energy bands E(qx,qy).

When I want to calculate the BC, the formula is $$\Omega_i(q)=i\left(\langle \partial_{qx}u_i|\partial_{qy}u_i \rangle-\langle \partial_{qy}u_i|\partial_{qx}u_i \rangle\right)$$

I am kind of confused as to what $u_i$ I should be using. I had just been using $u_i=E_i$, so taking the derivatives of the energy. Should I be using the eigenvectors. If I assume bloch waves, then I know there is the $e^{iqr}$ factor, but I want the derivatives in k space, so I am just not even sure what some pseudo code would look like.

As an aside, I am following this thesis which does the Harper model for a flux of 1/5.

So the prodedure is:

for each qx,qy:

  • Find E(q), $\vec{u(q)}$ from here I just dont know what to use for u.
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The question is you have a force given by the expression $ \mathbf F=-K(y\hat i+x\hat j)$, find the the work done in moving a particle from origin $(0,0)$ to $(a,0)$ to $(a,a)$. I tried finding the scalar potential. I equated $ \frac{\text dV}{\text dx}\hat i$ to $y\hat i$ and $ \frac{\text dV} {\text dy}\hat j$ to $x\hat j$ and found the potential function as $2xy$, but I keep getting the wrong answer. If anyone could point out my mistake, I would be thankful

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I am working on a mechanical system where I need each torque in each axis of a object for the euler lagrange equations. Sequence of rotations is Y-Z-Y. Object is in three dimensions.

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I m a noob, I wanted to figure out the time dilation for an observer on earth who passed 50,000 years, and the Traveler at light speed(99.9%) passed ONE DAY. I mean we pass 50,000 years on earth and the Traveler spends One day.Problem is I tried many online calculator, but each one is giving me different result with the same inputs.I m confused.I want to know the result. I also want to consider the LENGTH CONTRACTION/DISTANCE DILATION while calculating this.I m not very good at maths. Can somebody figure this out for me.

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I'm reading a book introducing gravity and find something I don't understand. Please check the attached image. The sentence in red bracket claims that if one adding up gravitational fields of infinitely many objects, the total gravitational field depends on the order in which we add them.

I'm just wondering how do summation order affect the total gravitational field? Is this some mathematical magic in infinite series?

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Big question here, and I'm willing to pay $ for a great answer.

I'm looking for someone to help me solve a riddle:

My company designed a window sheer with proprietary fabric with claims to better retain the indoor temperature by:

1) Blocking more heat/cold from the window than the average window share.<- we have tested and confirmed this.

2) ...and hopefully discovering we can regulate additional heat by blocking more IR waves from entering a room through a window vs a standard window sheer.

The labs I have spoken with suggested the ASTM E903 test to measure the amount of light (at various wave lengths) that pass through the sheer.

The goal would be to test both our sheer and a competitors sheer, then taking the results and calculated the heating cost savings a consumer would realize if they used our patented sheer vs a standard sheer.

I think the questions are: 1) Is it possible to arrive at such a conclusion 2) If so, are you willing to help solve the problem?

I appreciate the help and I'm happy to pay a consulting fee.

Thanks! Andrew

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Air can contain a certain amount of water vapor molecules.

I want to know that which any other non-hazardous gas can adsorb very more amount of water vapor molecules?

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Preliminaries: Consider the homogenous Maxwell's equations

$$\partial_\mu F^{\mu\nu}=0.$$

and

$$\partial_{\sigma} F_{\mu \nu}+\partial_{\mu} F_{\nu \sigma}+\partial_{\nu} F_{\sigma \mu}=0$$

Since $F^{\mu\nu}$ is antisymmetric there are a total of $\frac{16-4}{2} =6$ DOFs, all of which are physical.

Now, let us introduce a gauge potential $A_\mu$ so that

$$ F_{\mu\nu} = \partial_\mu A_\nu - \partial_\nu A_\mu $$

which also has 6 DOFs, and also leave the Maxwell equations invariant under a suitable transformation of $A_\mu$. However, we regard some of them as non-physical.

Question: Is my counting of degrees of freedom correct? If not, then where do these "extra" number of degrees arise in introducing the gauge potential $A_\mu$? If my counting is correct, then why are there redundant DOFs in $\partial_{[\mu}A_{\nu]}$ but not $F_{\mu\nu}$?

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