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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

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...

## \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

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 :)

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 \} =...

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.

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 >=...

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...

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...

Ahh yes! So with the minus sign, does this look right? Thanks

##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...

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...

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

Can you solve for N in these, or just plug in values of N?

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

I think the 'simplest formula' is meant to be ##NlnN-N## in this case

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...

I did- was a typo though. Thank you for your help! Very much appreciated

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...

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##

no, but i probably should have!

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

No, it was the next bit that I struggled with

Thank you you spotting that. That was just a typo. I do get that factor of d with my method

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...

Thank you. I wondered where that minus sign came from. Do you happen to know how to do the next part?

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...

It does help- thank you very much!

##\frac{V}{l}##

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

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

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

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...

Oops sorry its meant to be ##exp(3a) + 5##

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

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

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}##

##e^{3\lambda}##?

They are the exponentials of the eigenvalues of A

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...

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...

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

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.

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

Sorry, have I? There is 2Q on one plate and -Q on the other

The E field would still be constant if ##\sigma_2## is not equal to ##\sigma_3## ?

No, that's right. There's 2Q on one plate and -Q on the other plate

Why should the two inner plates be equal and opposite?

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