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    Combinatorics question -- Probability of being dealt a certain hand from a deck of cards

    Thank you! I considered this possibility, but discarded it on the following logic: If the question means that the sequence has to be all of the same suit, it should still be a factor of 4 in answer (C), as there are four possible suits for it to work with. If the question wants a specific...
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    Combinatorics question -- Probability of being dealt a certain hand from a deck of cards

    A quick translation of the question: We have a deck of 40 cards, containing four suits (hearts, diamonds, spades, clubs), in which each suit has an ace, a queen, a jack, a king and six number cards (2 to 7). From the deck, six cards are distributed randomly and successively to a player who picks...
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    I Derivative using the chain rule

    Thanks to both of you...I understand it now.
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    I Derivative using the chain rule

    Ok, thanks, I see where my mistake came in. So now I have ## \frac{d\sqrt{u}}{du}=(1/2)u^{-1/2} ## and ## \frac{du}{d\epsilon}= 2(y' + \epsilon g') ## so ## \frac {df}{d\epsilon} = (1/2)u^{-1/2}\times 2(y' + \epsilon g') ## but I seem to be a factor of ##g'## out.
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    I Derivative using the chain rule

    I'm coming back to maths (calculus of variations) after a long hiatus, and am a little rusty. I can't remember how to do the following derivative: ## \frac{d}{d\epsilon}\left(\sqrt{1 + (y' + \epsilon g')^2}\right) ## where ##y, g## are functions of ##x## I know I should substitute say ##u = 1...
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    A How do I define Open cluster membership?

    Sorry - I think I wasn't clear in my original post. I'm not looking for the definition of an open cluster, which you've summarised well, but I'm trying to define* membership of that cluster. I am studying some of the properties of a few open clusters, as identified by other authors using Gaia...
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    A How do I define Open cluster membership?

    I'm doing a study using Gaia DR2 data, and in particular trying to analyse the photometric data for groups of stars which some authors believe to be loosely gravitationally bound in open clusters. I am trying to find out a reasonable way of establishing cluster membership, before analysing the...
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    Direction of propagation of an EM wave

    Ah, well I can see that the term in the exponential brackets must be constant in order for there to be no attenuation over time, so I imagine that y must increase as t increases...hence propagation in the y-direction. So far so good. So I guess at z=0 the expression is zero, and this indicates...
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    Direction of propagation of an EM wave

    I'm reading my course book on ELectromagnetism and it is talking about a wave moving in the y-z plane but with polarisation in the x-direction, and it says that the equation $$\mathbf{E}=E_0 2i sin(k_0 z\ cos \theta) exp[i(k_0 y\ sin \theta - \omega t)]\mathbf{e}_x$$ Shows that there is no...
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    Object falling from rest, solution of differential equation

    Thanks a million. It's been a while since I've done this, and I seemed to be making a mistake at each stage...
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    Object falling from rest, solution of differential equation

    Ok, I take your point...I just thought maybe people had thought I was referring to v(x) which also gets used in kinematics quite a lot. The problem I had was really how to separate the variables. Anyway, having spent most of the day on this (!) I can now see how the separation of variables is...
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    Object falling from rest, solution of differential equation

    Still not right: I'm getting $$x(t)=\left(C+Dt\right)e^{-\frac{k}{m}t}+\frac{mg}{k}t$$ So taking out C, as I'll define the object to be at the origin at t=0, and differentiating: $$v(t)=e^{-\frac{k}{m}t}D\left(1-\frac{k}{m}t\right)+\frac{mg}{k}$$ Using initial conditions (object at rest)...
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    Object falling from rest, solution of differential equation

    Ahhh, thanks. That makes sense now. So $$\lambda_1=\lambda_2=-\frac{k}{m}$$ so $$x(t)=\left(C+Dt\right)e^{-\frac{k}{m}t}$$ If I include the fact that I made a mistake with the particular integral (should have been mg/k, thanks Lawrence) I might be nearly there. EDIT: I put the wrong...
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    Object falling from rest, solution of differential equation

    Thanks for both of these replies. Firstly, to Marioeden: Yes, that's the separation of variables technique that I'm familiar with. I usually look to separate the variables so that they are on either sides of the equation, then perform the integration using the chain rule as you mentioned...
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    Object falling from rest, solution of differential equation

    Homework Statement An object is falling from rest with air resistance modelled by $$kv_x$$. Where v_x is the object's velocity in the x-direction (downwards). Find an expression for the speed of the object as a function of time. Homework Equations $$mg-kv_x=m\frac{dv_x}{dt}$$ The...
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    Free and bound charge in a dielectric

    Homework Statement I'm just trying to understand better what happens at the interface between a conductor and a dielectric, particularly with regard to free and bound charge. I would like to know: - under what conditions can a dielectric acquire free charge, and how this free charge...
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    Checking method for calculating free charge from Electric field in LIH

    I know that the potential is zero at the surface because I have the expression for V(r) and it works out as zero on the surface. I've just realised, though, that I could choose an arbitrary Gaussian surface in the dielectric and use that to calculate the free charge per unit length. Would...
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    Checking method for calculating free charge from Electric field in LIH

    Homework Statement I'm asked to find the free charge per unit length on a cylinder which is surrounded by a LIH dielectric material. I have an expression for the electrostatic potential $$V(r)$$ in cylindrical coordinates. Homework Equations $$\mathbf{D}=\varepsilon \varepsilon_0...
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    Congruence identities using Fermat's Little Theorem

    Thanks for taking the time out. I feel like I'm probably not far away! $$9^2\equiv13\ (mod\ 17)$$ So I guess I can consider $$9^{11}=\left(9^2\right)^5\times9$$ and simplify to: $$43^{43}\equiv13^5\times 9$$ But still unsure how to proceed. I'll have another think...
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    Congruence identities using Fermat's Little Theorem

    Homework Statement Show the remainder when 43^43 is divided by 17. Homework Equations $$43 = 16 \times 2 + 11$$ $$a^{p-1}\equiv1\ (mod\ p)$$ The Attempt at a Solution I believe I can state at the outset that as: $$43\equiv9\ (mod\ 17)$$ Then $$43^{43}\equiv9^{43}\ (mod\ 17)$$ and that I...
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    Finding the range of validity of a taylor series

    Homework Statement I have to give the range of validity for a Taylor series built from an expression of the form: (1+(a/b)x)^c Homework Equations The Attempt at a Solution Obviously the validity does not extend to x=-(b/a) on the negative side, but should I then state that...
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    Vector transformation, cylindrical to Cartesian

    Homework Statement I have a result which is in the form (cylindrical coordinates): $$ A\boldsymbol{e_{\theta }}=kr\boldsymbol{e_{\theta }} $$ And I have to provide the answer in cartesian coordinates. Homework Equations I know that the unit vectors: $$ \boldsymbol{\hat{\theta}...
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    Limits of an expression in x

    Thanks, very clear.
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    Limits of an expression in x

    When sketching a graph I'm told to assume that the expression: f(x) =( e^x)/x Tends towards the infinite as x tends towards the infinite. Can someone show me how to check this? Thanks
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    Electron drift speed in metals

    Is there a physical explanation for why electrons move through a metal with a relatively low drift speed? Or is it just the observed phenomenon? I find it hard to visualise electrons moving relatively slowly through a metal despite the current flowing through it being quick...can anyone help? Thx
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    Infinite plane of charge

    Thanks for all those replies...that's much clearer.
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    Magnetic flux through a closed surface

    Thanks. I had forgotten that the magnetic field lines can be thought of as passing through the inside of the magnet, so I take your point that they all cancel out. That brings up a new question though: if the North pole of a magnet has field lines coming in and going out of it, and so does...
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    Infinite plane of charge

    I'm a little confused by the textbook I'm working from, which asserts: "the electrical field from an infinite plane of charge does not decrease with distance" Why doesn't it decrease as the square of distance from the plane, from Coulomb's law? Thanks in advance
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    Question about Flux through a closed surface

    Thanks. So is it fair to say that if you create a closed surface around a single pole of a magnet, you still have field lines coming in and going out...meaning that the single pole is actually a dipole? In that case, I'm wondering whether splitting a magnet up into two poles is actually...
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    Question about Flux through a closed surface

    Can you set up a closed surface around one of the poles of a magnet? The problem I have is that it seems to violate the principle that the magnetic flux through a closed surface is always zero. Thanks
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