# Search results for query: *

1. ### How do you show |r_1-r_2| is rotationally invariant

But is this not the transformation one makes in lagrangian mechanics to then say ##\frac{\partial L}{\partial \dot{q}} (n \times r) = constant ## by Noether's theorem? Many thanks
2. ### How do you show |r_1-r_2| is rotationally invariant

Homework Statement How do you show |r_1-r_2| is rotationally invariant Homework Equations The Attempt at a Solution So i get that we need to show that it is invariant under the transformations ## r_1 \rightarrow r_1 + \epsilon (n \times r_1)## ## r_2 \rightarrow r_2 + \epsilon (n \times...
3. ### Generating Function for Lagrangian Invariant System

## \frac{\partial L}{\partial \dot{Q}} = \frac{\partial L}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial \dot{Q}} =\frac{\partial L}{\partial \dot{q}} \frac{1}{1+\dot{K}'} = p \frac{1}{1+\dot{K}'} ## ?? is this along the right lines? Many thanks
4. ### Generating Function for Lagrangian Invariant System

Ahh okay then, so I guess i can't make that assumption. In which case i don't know how to proceed, any tips? Thank you, i will change them :)
5. ### Generating Function for Lagrangian Invariant System

Homework Statement Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G. Homework Equations The Attempt at a Solution ##\delta q = \{ q,G \} =...
6. ### Measurement of Lx: Result of Measurement?

But how do i find the result of a measurement. I can show that the expected value is zero, but i don't know what the outcome of an individual measurement would be.
7. ### Measurement of Lx: Result of Measurement?

Homework Statement If we have a wave function ##\psi =zf(r)## and we take a measurement of ##L_x## what is the result of the measurement? Homework Equations The Attempt at a Solution So i know we can write ##L_x=\frac{1}{2}(L_+ + L_- )## and that ##|\psi > = g(r) |1,0> ## so ##L_x |\psi >=...
8. ### Interferometer- 2 spectral lines

So would we say that ##n=m+1## ##\delta_1=2k_1 d## ##\delta_2=2k_2 d## ##\delta_1-\delta_2=\pi = 4d(\frac{1}{\lambda_1}-\frac{1}{\lambda_2})## ##d=\frac{\lambda_1 \lambda_2}{4 \Delta \lambda}## I have a few issues with doing this 1) i don't know why ##\delta_1-\delta_2=\pi## and 2) when i plot...
9. ### Interferometer- 2 spectral lines

Homework Statement An interferometer is illuminated by light from a sodium lamp, which emits two narrow spectral lines at wavelengths of 589.0nm and 589.6nm, with the intensity of the 589.0nm line being twice that of the 589.6nm line. Show that there are values of d at which the visibility of...
10. ### Heat transfer to a water bath

Ahh yes! So with the minus sign, does this look right? Thanks
11. ### Heat transfer to a water bath

##V=\frac{M}{\rho}## ##\Delta V= \delta \big{(} \frac{1}{\rho_l} + \frac{1}{\rho_s} \big{)}## where ##\delta## is the mass that is converted from solid to liquid. ## \delta= 9.58 \times 10^{-3}## Can we then use ##lM=T\Delta S = \Delta Q## ##\Delta Q = l \delta = 3191.25 J## ? where l is the...
12. ### Heat transfer to a water bath

Homework Statement A cylinder is fitted with a piston and is in thermal contact with a heat bath at 273K. Initially the volume in the cylinder is filled with 10kg of pure H2O and about half of this is liquid and the other half is solid. The piston is lowered so as to reduce the volume by 2 ×...

Homework Statement The emission of radiation from the Sun’s disc is observed to peak at 0.5 μm wave- length and that from the Moon’s disc at 10.0μm. A heat engine to power a Moon base is to be constructed using radiation collected from the Sun. What is the maximum theoretical efficiency of such...
14. ### How can gyroscopes be used to find the angular velocity?

I was reading that gyroscopes can be used to measure the angular velocity of precession, such as in the Hubble space telescope, but mathematically how can this be done? Many thanks
15. ### Accuracy of Stirlings Formula

Can you solve for N in these, or just plug in values of N?
16. ### Accuracy of Stirlings Formula

Okay, but how would i go about working out for which N gives an answer to within 2% of the true value? Thanks :)
17. ### Accuracy of Stirlings Formula

I think the 'simplest formula' is meant to be ##NlnN-N## in this case
18. ### Accuracy of Stirlings Formula

Homework Statement How big must N be for the simple version of stirlings formula to be accurate to within 2% Homework Equations The Attempt at a Solution So I think the starting point is ##\frac{N lnN-N}{lnN!} =\alpha ## where ##\alpha=0.98## But i have no idea how to solve this expression...
19. ### Calculate the temperature of neutrons emerging from a reactor

I did- was a typo though. Thank you for your help! Very much appreciated
20. ### Dielectric problem -- two line charges inside a dielectric cylinder

We know that V-> 0 as r-> ##\infinity## inside the dielectric so the potential here must take the form ##V=\sum{r^{-n}(c_nsin(n\phi)+d_ncos(n\phi)}## inside the cavity we have ##V=a_0+b_0ln(r)+\sum{(a_nr^n+b_nr^{-n})(c_nsin(n\phi)+d_ncos(n\phi))}## Am i right in assuming that we can say that V...
21. ### Calculate the temperature of neutrons emerging from a reactor

So can we approximate this as ##(4000)^3e^{-\frac{4000}{v_{th}^2}}\int_{4000}^{4010}{dv}=2(2000)^3e^{-\frac{2000}{v_{th}^2}}\int_{2000}^{2010}{dv}## ? ##T=\frac{m}{K_B}\Big(\frac{4000^2-2000^2}{\ln{\frac{4000^3}{2(2000^3)}}}\Big)^2##
22. ### Calculate the temperature of neutrons emerging from a reactor

no, but i probably should have!
23. ### Calculate the temperature of neutrons emerging from a reactor

Sorry that is meant to be ##v_{th}^2## I have missed of the ##^2##. That notation is meant to be putting in limits from 4000 to 4010
24. ### Dielectric problem -- two line charges inside a dielectric cylinder

No, it was the next bit that I struggled with
25. ### Dielectric problem -- two line charges inside a dielectric cylinder

Thank you you spotting that. That was just a typo. I do get that factor of d with my method
26. ### Calculate the temperature of neutrons emerging from a reactor

Homework Statement A collimated beam of thermal neutrons emerges from a nuclear reactor and passes through a speed selector into a detector. The number of neutrons detected in a second with speeds in the range 4000 to 4010 m s−1 is twice as large as the number per second detected with speeds in...
27. ### Dielectric problem -- two line charges inside a dielectric cylinder

Thank you. I wondered where that minus sign came from. Do you happen to know how to do the next part?
28. ### Dielectric problem -- two line charges inside a dielectric cylinder

Homework Statement Consider a cylindrical hole of radius a and infinite length cut into a dielectric medium with relative electric permittivity ε (the interior can be treated as a vacuum). Inside the hole there are two line charges of infinite length with line charge densities λ and −λ...
29. ### Poynting vector of current carrying wire

It does help- thank you very much!
30. ### Poynting vector of current carrying wire

##\frac{V}{l}##
31. ### Poynting vector of current carrying wire

Inside the wire we have a non-neutral charge. Negative electrons flow in one direction, leaving positive ions behind. Hence we have an E-field in the direction of current? Not sure how to calculate its magnitude though
32. ### Poynting vector of current carrying wire

It is neutral. However I know there needs to be a poynting vector so I don't know how this works
33. ### Thermal diffusion equation

I don't understand. We don't have angular dependence here so surely we can have non linear dependence on r. If you work through the algebra I can't see how I could not include this term. I'm not too sure how to use this without knowing how to calculate q
34. ### Thermal diffusion equation

Homework Statement Gas with thermal conductivity κ fills the space between two coaxial cylinders (inner cylinder radius a, outer cylinder inner radius b). A current I is passed through the inner cylinder, which has resistivity ρ. Derive an expression for the equilibrium temperature of the inner...
35. ### Eigenvectors and eigenvalues

Oops sorry its meant to be ##exp(3a) + 5##
36. ### Eigenvectors and eigenvalues

Will have eigenvalues ##e^{3a+5}## with the same eigenvectors Thank you for your help
37. ### Eigenvectors and eigenvalues

Okay. But how do you find the eigenvalues of ##exp(3A)+5I##?
38. ### Eigenvectors and eigenvalues

the eigenvalues of ##exp(B)## are ##e^b## but ##b=3a## where a are the eigenvalues of A for ##B=3A##. Hence the eigenvalues are ##e^{3a}##
39. ### Eigenvectors and eigenvalues

##e^{3\lambda}##?
40. ### Eigenvectors and eigenvalues

They are the exponentials of the eigenvalues of A
41. ### Eigenvectors and eigenvalues

Homework Statement Find the eigenvalues and eigenvectors of the matrix ##A=\matrix{{2, 0, -1}\\{0, 2, -1}\\{-1, -1, 3} }## What are the eigenvalues and eigenvectors of the matrix B = exp(3A) + 5I, where I is the identity matrix? Homework Equations The Attempt at a Solution So I've found...
42. ### Poynting vector of current carrying wire

Homework Statement A long straight wire of radius a and resistance per unit length R carries a constant current I. Find the Poynting vector N = E × H at the surface of the wire and give a sketch showing the directions of the current, the electric field E, the magnetic field H, and N. Integrate...
43. ### Charge density of capacitor plates

I'll use surfaces from now on :) I still don't see why you wouldn't have a uniform field with different charges on the inner surfaces
44. ### Charge density of capacitor plates

Ahh I see the confusion now. I think then maybe ##\sigma_4=\frac{Q}{2A}## I have define plates 1234 to be the sequence of planes met if we travel from the top of the capacitor to the bottom where the top capacitor plate has charge 2Q and the bottom charge -Q.
45. ### Charge density of capacitor plates

Sorry if I am being really stupid but having ##\sigma_2## not equal to ##sigma_3## gives a uniform field in between the plates of strength ##\frac{\sigma_2}{2\epsilon_0 } + \frac{\sigma_3}{2\epsilon_0}## using gauss' law
46. ### Charge density of capacitor plates

Sorry, have I? There is 2Q on one plate and -Q on the other
47. ### Charge density of capacitor plates

The E field would still be constant if ##\sigma_2## is not equal to ##\sigma_3## ?
48. ### Charge density of capacitor plates

No, that's right. There's 2Q on one plate and -Q on the other plate
49. ### Charge density of capacitor plates

Why should the two inner plates be equal and opposite?
50. ### Laplacian for hyperbolic plates

I don't see how you could sketch it without working out ##\sigma## Could we not have a non zero volume charge though? To satisfy the conservation of charge