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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

    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?

    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

    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

    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

    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

    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

    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

    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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    Magnetic flux through a closed surface

    This is always zero, right? What if you construct a closed surface which only encompasses one of the poles of a magnet? Surely there would then be a non-zero flux as the inside of the surface would constitute a source (or sink) of magnetic field lines. I'm new to electromagnetism, so any...
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    Diffusion equation, boundary conditions

    EDIT: The subscripts in this question should all be superscripts! Homework Statement I'm trying to solve a temperature problem involving the diffusion equation, which has led me to the expression: X(x) = Cekx+De-kx Where U(x,y) = X(x)Y(y) and I am ignoring any expressions where...
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    Zigzag carbon nanotubes

    Homework Statement I'm trying to express the conditions for a zigzag carbon nanotube in a mathematical vector. Homework Equations B = na1 + ma2 The Attempt at a Solution I want to say "either a=0 or b=0", but must I also say that it can't be a=b=0 or is that already excluded in...
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    Integration by substitution (I think)

    Homework Statement Integral of d.cos j with regard to d.sin j Where d is a constant. Homework Equations The Attempt at a Solution I don't know how to approach this. I can substitute u=d.sin j Then I have Integral of dz/dj with regard to dz, but not sure where to go from here. Any help...
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    Moment of inertia question

    If you have a roundabout spinning with a man standing on it close to the centre, and then he walks out towards the edge of the roundabout, angular momentum is conserved, but kinetic energy is not (the roundabout rotates with a smaller angular speed). I'd like to know where the kinetic energy in...
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    Damped harmonic oscillator being forced

    Homework Statement A damped harmonic oscillator is being forced. I have to say whether it is direct forcing or forcing by displacement. I have the equation of motion which is expressed in terms of the particle's height above the equilibrium point and an expression for the force being...
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    Differential equation of motion

    Homework Statement I have to set up a differential equation of motion for a particle undergoing projectile motion, for which I know the initial position and velocity vectors Homework Equations The kinematics equations. The Attempt at a Solution Well I can easily take a(t) = -gj...
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    Integration problem

    Homework Statement I'm trying to do this problem from S. P. Thomson's Calculus Made Easy (see attached image file) Homework Equations The Attempt at a Solution I've produce the answer here, but it seems to be different from the mathcad solution, which is...
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    Region of motion

    Homework Statement Find the region of motion of a particle in a system following a potential energy function of: U(x)=ax4+bx3+cx + d and total energy of E=3 I know the values of a, b, c and d. Homework Equations Etotal=U(x)+Ekin The Attempt at a Solution I know that I can the turning points...
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    Differentiation (basic)

    Homework Statement d(x(t)2)/dt Homework Equations The Attempt at a Solution I guess that this should be: 2x(dx/dt) but I'm not sure how to justify it: u= x d(u2)/du = 2u (d(u2)/du)(du/dt) = d(u2)/dt So 2u(du/dt) = 2x(dx/dt) Is this right?
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    Integration problem

    Homework Statement v(dv/dx)= k(2v+1) Find an explicit solution for x in terms of v. Homework Equations v=speed, x=distance, k=constant The Attempt at a Solution I can solve this by dividing by (2v+1), then integrating with respect to x using the chain rule. I believe there is...
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    Notation for domains

    If you have an equation with a variable which isn't defined for a given value or values, how do you express this in proper notation? For example: x=1/((y-2)(y-3)) Do I write simply " y<2 or 2<y<3 or 3<y" or is there a better way to express it? Thx
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    Numerical value of complex number

    Homework Statement I need to understand why \left|4+i \right|=4.123 and why this is shown by: \sqrt{4^{2}+1^{2}}=4.123 Homework Equations i^{2}=-1 The Attempt at a Solution If I find the square root of this expression squared, then I come up with \sqrt{16-1+8i} which is...
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    Rearranging an expression in sqrt argument

    Homework Statement Rearranging an equation...can't quite see how it's done. Homework Equations r=\sqrt{2.5^{2}cos^{2}(t/2)+5^{2}sin^{2}(t/2)} r=2.5\sqrt{cos^{2}(t/2)+4sin^{2}(t/2)} r=2.5\sqrt{1+3sin^{2}(t/2)} The Attempt at a Solution I know that cos^{2}(x)+sin^{2}(x)=1 but...
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    Integration trouble

    I'm just playing around with a linear air resistance model to show that a marble, diameter 2cm, mass 13g, takes 3.96 seconds to fall 77m off a bridge to the water. I've done it by starting with ma = mg - c1Dv Where m is mass, D is diameter, v is velocity downwards and a is acceleration...
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    Rearranging an equation

    I've got a kinematics equation modelling the flight of a stone: v = 10-gt s= -1/2 g t^2 +10t +2 I can't remember how to get a value for v which doesn't contain t. I tried rearranging the first equation and introducing it to replace t in the second, but can't seem to get v isolated...
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    Maximum wavelength of EM

    Is there any lower limit on the frequency of electromagnetic radiation? I imagine that beyond a certain frequency it becomes impossible to detect, but is there anything in the maxwell equations that establishes a maximum wavelength (besides the limits of the size of the universe)? Thanks