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Maybe it was too late for me to be posting before but my first method misses the extra constant you'd get.

That's definitely a good idea! Integrating c_v/T we find s = \frac{AT^3}{3} +f(v) for some f. Then with the Maxwell relation f'(v) = \frac{B'(T)}{v-v_0} which means that B'(T) = E for some E. So this confirms what I found before (much more easily).

Homework Statement The constant-volume heat capacity of a particular simple system is c_v = AT^3 where A is a constant. In addition the equation of state is known to be of the form (v-v_0)p = B(T) where B(T) is an unspecified function of T. Evaluate the permissible functional form of B(T)...

So in my answer I found that the compressibility is infinite if \mu, T is fixed which has to be the case if \mu is determined by T and p only. A constant T and \mu] implies a constant p irrespective of V. I'm just not sure if that is the answer they want and what the physical significance of it...

I feel silly now as I wrote d\mu =-sdT +vdp . If the system is equilibrium should not dU + dU_r = 0 = (T-T_r)dS + (\mu-\mu_r)dN mean that T=T_r and \mu_r = \mu if the system undergoes a quasi static change? I guess that would 'explain' why I found the derivative to be infinite as the...

Thank you for the response. I agree with you on your first point but I do not understand why the chemical potential matches the reservoir only if the pressure is constant. I thought the condition for equilibrium would set the chemical potential to be the reservoir's regardless, and should not...

Homework Statement A cylinder is fitted with a piston, and the cylinder contains helium gas. The sides of the cylinder are adiabatic, impermeable, and rigid, but the bottom of the cylinder is thermally conductive, permeable to helium, and rigid. Through this permeable wall the system is in...

Thanks! I notice that if in the denominator the as and b are ignored, then my final equation matches yours.

Ok this time I expanded around V/2 like you said :P and got V_1 = V/2 + \frac{4(a_2-a_1)}{\frac{16(a_2+a_1)}{V} +\frac{2RTV^2}{(V/2-b)^2}}

Thanks for the response. First I should say I didn't mean to write the N as it is 1, although it doesn't really matter. I think I understand what you mean, but I am apprehensive about the right hand side expansion, unless you mean something like \frac{RT}{V_1} + \frac{RTb}{V_1^2} -...

Homework Statement Two ideal van der Waals fluids are contained in a cylinder, separated by an internal moveable piston. There is one mole of each fluid, and the two fluids have the same values of the van der Waals constants b and c; the respective values of the van der Waals constant ''a'' are...

Can you write it with in LaTeX?

Homework Statement The fundamental equation of a system of \tidle{N} atoms each of which can exist an atomic state with energy e_u or in atomic state e_d (and in no other state) is F= - \tilde{N} k_B T \log ( e^{-\beta e_u} + e^{-\beta e_d} ) Here k_B is Boltzmann's constant \beta = 1/k_BT...

I see my problem: there are two Qs. The first equation was the heat from the subsystem and the energy and entropy the heat RHS.

Homework Statement Assume that one mole of an ideal van der Waals fluid is expanded isothermally, at temperature T_h from an initial volume V_i to a final volume V_f. A thermal reselvoir at temperature T_c is available. Apply dW_{RWS} = \left ( 1 - \frac{T_{RHS}}{T} \right ) (-dQ) +(-dW) to a...

This is the beginning of writing it out as two derivatives, the physicist's way. Simply divide and multiply each by \delta \phi and \delta \phi^* respectively and take the infinitesimal limit.

That sounds reasonable. The wording of question doesn't imply you have some freedom in choosing your parameters, but since the number of equations is under determined I would say you are likely right.

Well that's what I did, hence z = r \cos \theta , however to me that didn't give an obvious solution.

But I'm also not sure how you can get 3 parameters from one equation. It is strange that it says for any n=2 state though, when at least if l=1, m=0, you got 0 as the probability. Also perhaps that the electric field is weak needs to be employed.

Ah ok. It's just I would have thought you take the inner product of the potential, that is, the potential is your operator.

Why would you not just be able to do \frac{\partial V}{\partial \phi} = V'(\phi^* \phi) \phi^*? Also, shouldn't it be V( \phi \phi^* + \phi \delta \phi^* + \phi^* \delta \phi + \delta \phi \delta \phi^*)-V(\phi \phi^*) =V( \phi \phi^* + \phi \delta \phi^* + \phi^* \delta \phi + \delta \phi...

Not that this is that much help but shouldn't the potential be proportional to r \cos \theta not r? With this you get zero for m=-1, but the same otherwise.

Here's what I think is one way. Consider \nabla_\mu W_\nu = \partial_\mu W_\nu -\Gamma^s_{\nu \mu} W_s = \partial_\mu W_\nu -\tfrac12 W_s g^{s \sigma} ( \partial_\mu g_{\sigma \nu } + \partial_\nu g_{\sigma \mu} - \partial_\sigma g_{\nu \mu}) = \partial_\mu W_\nu -\tfrac12 W^\sigma (...

I just can't imagine a situation where if <p|L_n|p'>=0 for all |p> and |p'> and they aren't equal that L_n \neq 0 . For example say |p'> = L_n|p> . I mean it must be some state right? Then the condition implies <p'|p'> =0 which by the properties of vector implies |p'> = L_n |p> =0.

Thank you for your help. I've found similar papers and although I don't know if the authors in this paper did this (my equations still don't completely match), there are ways to get the method to work. If you use f'(R) R - 2 f(R) +3 \Box f'(R)= 0 to replace the f(R) term with so that all the...

So I don't understand some of your notation. Do you mean L_n^\dagger = L_{-n}? Also are |p> and |p'> arbitrary? It seems like having an operator act on a ket and then taking the inner product always giving zero would mean the operator gives zero on the ket. All the extra parts about n aren't...

I don't think I understand what you mean. So the contracted equation is f'(R) R - 2 f(R) +3 \Box f'(R)= 0 which when using f(r) not f(R) in my code I get their equation 7. I mean to write the components don't you need to do \nabla_\mu \nabla_r f'(R) = \nabla_\mu ( f''(R) \partial_r R )...

But don't you have to do this in the right hand side of their equation 2? And their components had f''' and f'' terms; wouldn't they have to come from this differentiation?

Sorry I used the code for something else and forgot to change the way the metric was written. I have corrected it.

I am able to obtain the contraction. Considering how many terms there are I really do not believe any of this is a coincidence. 4 and 5 are the same except a few terms have the wrong sign. I think equation 6 had an extra term for me. Now the Ricci scalar I got was \frac{A'(r) B'(r)}{2 A(r)^2...

So to obtain equations 4 and 7 (or something close to them) on Mathematica I had to use use the second form of the equation I gave. Now, this could be because I implemented it incorrectly on Mathematica, however when I used R and not r my equations were far more complicated as I said, so I'm not...

In this paper (https://arxiv.org/abs/astro-ph/0603302) the authors derive the field equations for f(R) gravity considering a spherically symmetric and static metric. Now the Ricci scalar only depends on r so you could write f(R(r)) = g(r) for some g. However what it seems the authors have done...

Homework Statement Suppose a magnetic monopole q_m passes through a resistanceless loop of wire with self-inductance L. What current is induced in the loop? Homework Equations \nabla \times \textbf{E} = - \mu_0 \textbf{J}_m - \frac{\partial \textbf{B}}{\partial t} \nabla \cdot \textbf{B} =...

I'm not sure what you mean, but I found the answer they were looking for. From a \ddot{\theta} = 2kx/M I found a \theta and then added it to x from which we get a term corresponding to a constant acceleration of 2mg/(M+2m) and an oscillation with the same frequency but with amplitude as given...

If I find the acceleration a \ddot{\theta} perhaps then combining the two it will be obvious.

I did combine two questions here. The previous question was without the string being inelastic and the constant acceleration term was a \ddot{\theta}. I guess there is no other acceleration with the coordinates I have chosen.

I've looked at it a bit and I'm not sure what the sign error would be.

Homework Statement A uniform cylindrical drum of mass M and radius a is free to rotate about its axis, which i is horizontal. An elastic cable of negligible mass and length l is wrapped around the drum and carries on its free end a mass m. The cable has elastic potential energy \tfrac12...

Thanks I think I had a mental block because of the index notation.

This is from a general relativity book but I think this is the appropriate location. The proof that \nabla_{[a} {R_{bc]d}}^e=0 is as follows: Choose coordinates such that \Gamma^a_{bc}=0 at an event. We have \nabla_a {R_{bcd}}^e = \partial_a \partial_b \Gamma^e_{cd} - \partial_a...

So the temporal component is the first equation and the spatial the second? And yes the author does show that this reduces to the Euler and continuity equation for speeds much less than that of light.

Hi thanks for you answer it is quite helpful. In your last step when you contract the second term, how did that work? Edit: thanks I figured it out.

In Woodhouse's 'General Relativity' he finds an expression for the energy-momentum tensor of an isotropic fluid. If W^a is the rest-velocity of the fluid and \rho is the rest density then the tensor can be written as T^{ab} = \rho W^aW^b - p(g^{ab} -W^aW^b) for a scalar field p. The...

I made a mistake in copying this out. It should be ''plane harmonic waves of frequencies.''

Sorry I didn't notice your question. Here \textrm{r}_1 and \textrm{r}_2 are the positions of the comet and Earth, respectively.

The previous question was from chapter 4 of the book and had a section on orbits from inverse square law forces. It had subsections on hyperbolic orbits and the deflection, from which I was able to obtain the previous result.

Also the formula for \dot{\textrm{r}}_1 is \dot{\textrm{R}} + m_2/M \dot{\textrm{r}} and we find that the total momentum is m_1 \dot{\textrm{r}}_1 + m_2 \dot{\textrm{r}}_2 = M \dot{\textrm{R}} so that momentum is conserved according to them.

What about the equation \mu \ddot{\textrm{r}} = \textrm{F}? Wouldn't \dot{\textrm{r}} change in this case? Also the deflection angle is the one you calculate from the previous question, which is confirmed in the solutions.

OK I uploaded a couple of images. The first shows the set up and the second what has happened in the reference frame of the Earth. Here \dot{\textrm{R}} is \frac{m_1 \dot{\textrm{r}}_1 + m_2 \dot{\textrm{r}}_2}{m_1+m_2} which is the velocity of the centre of mass and \dot{\textrm{r}} =...

Homework Statement Suppose the asteroid of [other problem] has a mass of 6 \times 10^{20} \textrm{kg} . Find the proportional change in the kinetic energy of the Earth in this encounter. What is the change in the semi-major axis of the Earth's orbit? By how much is its orbital period...