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    I Stuck on the probability of rolling 'p' with 'n' s-sided dice

    Hi, I've been following the derivation of wolfram mathworld for this problem and I'm running into some trouble regarding the summation indices. Currently I am at the step where we have found that (it's pretty much just binomial expansion and taylor series to get to this point) $$ f(x) =...
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    Find B and H everywhere for a magnetized infinite cylinder

    Homework Statement An infinitely long cylinder of radius a has its axis along the z-direction. It has Magnetization ##M=M_0(s/a)^2\hat{\phi}## in cylindrical coordinates where ##M_0## is a constant and s is the perpendicular distance from the axis. Find the values of ##\vec{B}## and ##\vec{H}##...
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    Electric field inside a uniformly polarized cylinder

    Homework Statement This is problem 4.13 from Griffiths. A long cylinder of radius a carries a uniform polarization P perpendicular to its axis. Find the electric field inside the cylinder. Homework Equations ##\int \vec{E}\cdot dA = q_{encl}/\varepsilon_0## The Attempt at a Solution [/B] We...
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    Drawing a Timing Diagram

    Homework Statement Draw the diagram for the following circuit given the following conditions: 1) X=Y=Z=1 2)X=Y=1, Z=0 3)X=Y=0, Z=1 4)X=1, Y=Z=0 Homework Equations The Attempt at a Solution [/B] ##W=XZ'+YZ##, ##V=Y'Z+XY## 1) W = 0 + 1 = 1 V = 0 + 1 = 1 and now I'm not sure how to get the...
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    Calculating the surface charge of a sphere and a conducting shell

    Homework Statement (Problem 2.38 From Griffth's Electrodynamics): A metal sphere of radius R, carrying charge q, is surrounded by a thick concentric metal shell (inner radius a, outer radius b). The shell carries no net charge. Find the surface charge density ##\sigma## at R, a, and b...
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    I Why can we WLOG derive Simpson's rule over interval -1 to 1

    On the Simpson's Rule wikipedia page they mention in their derivation that the calculation can be simplified if one notices that there is no loss in generality in setting ##a=-1## and ##b=1## for the integral ##\int_{a}^{b}P(x)\cdot dx## as a result of scaling. I'm not entirely sure what...
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    Determining if a card is in the same pile as another card

    Homework Statement Write a predicate to determine if two cards are in the same pile. The placement of the cards is given as facts above(x,y), x is above y, or below(x,y), x is below y. I'm supposed to do this using Prolog which is a first-order logic language. Homework Equations The Attempt...
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    Optimal Path from source to multiple destinations

    Homework Statement You are placed in a 2-dimensional labyrinth at a starting location ##s## and must travel to n goal locations ##g_1, g_2, ..., g_n##. Determine the optimal path using the A* algorithm. g = goal cell, s = start cell example input: 3 11 //row length, column length # # # # # # #...
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    For a round-robin of n teams find the number of different outcomes

    Homework Statement In a round-robin tournament each team plays every other team once, find the number of different outcomes possible for ##n## teams. e.g. for 4 teams the possible outcomes are: |3-0 | 3-0 | 2-1 | 2-1 |...
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    Showing equivalent potential expressions for a Transverse String

    Homework Statement I'm going through the derivation here starting on page 16. This image adds some context: . Generalizing their result to the i'th particle they find the extended distance between two masses being ##\Delta l= \frac{(y_i-y_{i-1})^2}{2a}## Then since the potential energy is...
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    Proving a thermodynamic relationship

    Homework Statement Prove that ##TdS = C_vdT + \alpha T / \kappa dV## Homework Equations ##T dS = dU - pdV## ##\alpha = \frac{1}{v}\left(\frac{\partial v}{\partial T}\right )_P## ##\kappa = -\frac{1}{v}\left(\frac{\partial v}{\partial P}\right)_T## The Attempt at a Solution The ##C_vdT## part...
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    Moment of Inertia of a hollow cone about its base

    Homework Statement Show that the moment of inertia of a hollow cone of mass M, radius R, and height h about its base is ##\frac{1}{4}M(R^2+2h^2)## Homework Equations ##I=\int r^2dm## where r is the perpendicular distance from the axis Surface Area of a cone ##= \pi R (R^2+h^2)^{1/2}## The...
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    Changing Independent Variable in the Bessel Equation

    Homework Statement Given the bessel equation $$x^2\frac{d^2y}{dx^2} + x\frac{dy}{dx} -(1-x)y=0$$ show that when changing the variable to ##u = 2\sqrt{x}## the equation becomes $$u^2\frac{d^2y}{du^2}+u\frac{dy}{du}+(u^2-4)y = 0$$ Homework Equations The Attempt at a Solution ##u=2\sqrt{x}##...
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    Finding the Minimum Work Done for a Thermodynamic Process

    Homework Statement Given two indistinguishable objects at the same initial temperature ##T_i##, calculate the minimum work done by a refrigerator functioning between the two objects till one of the objects reaches a new temperature ##T_2##, assume constant heat capacities and constant pressure...
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    Finding the number of moles of an ideal gas in a capillary

    Homework Statement The temperature across the capillary with constant cross-sectional area and length L is given by ##T=T_0e^{-kx}##. Assuming an ideal gas and constant pressure show the number of moles to be: $$n=\frac{PV(e^{kL} - 1)}{RkLT_0}$$ Homework Equations ##PV=nRT## The Attempt at a...
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    Calculating work done on a gas

    Homework Statement n moles of an ideal gas are placed in a frictionless piston with weight ##w_p## and cross-sectional area ##A##. The quantity ##\gamma = \frac{c_p}{c_v}## is a constant, the gas is originally at equilibrium values##(P_i, V_i, \theta_i)## and the external pressure is taken to...
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    Falling Rain Drop (Variable Mass)

    Homework Statement Suppose a rain drop with mass ##m_0\neq 0## is falling due to gravity with initial velocity ##v_0##, assume ##\frac{dm}{dt}=k=##constant. Solve the differential equation and determine the velocity as ##t\to\infty## Homework Equations ##F=\frac{dp}{dt}=\dot{m}v+m\dot{v}## The...
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    Determining <r> for the hydrogen atom

    Homework Statement The equation for the normalized ##n=3##, ##l=2##, ##m=0## wavefunction is given by $$\psi_{320}=\frac{1}{81\sqrt{6\pi}}\left(\frac{1}{a_0}\right)^{3/2}\left(\frac{1}{a_0^2}\right)r^2e^{-\frac{r}{3a_0}}(3cos^2\theta-1)e^{i\phi}$$ Determine the expectation value ##<r>##...
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    Deriving Capacitor Voltage for a Discharging Capacitor

    Homework Statement I'm trying to derive the voltage waveform across the capacitor for a discharging capacitor that has been fully charged by a DC power supply ##v_0##, i.e. ##v_c(t=0)=v_0## and then at ##t=0## the switch is flipped and the capacitor starts to discharge. Homework Equations...
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    Thevenin's Theorem Problem

    Homework Statement Determine the voltage across ##R_3## in the following figure assuming an input voltage ##v_0## of 10V is applied across the open terminals. Let ##R_1=3\Omega##, ##R_2=15\Omega##, ##R_3=10\Omega##, ##R_4=5\Omega## and ##R_5=2\Omega## and Homework Equations The Attempt at...
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    Simplifying output for a XOR gate using Boolean Algebra

    Homework Statement I'm trying to show that the output of this XOR circuit is ##F=A'B+AB'##, Homework Equations ##(A+B)'=A'\cdot B'## ##(A\cdot B)'=A'+B'## The Attempt at a Solution From the gates the output is ##[(A\cdot B)+(A+B)']'##, using De Morgan's laws this becomes ##[(A\cdot...
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    Finding Internal Resistance of a Battery

    Homework Statement A real battery can be modeled as an ideal voltage source in series with a resistor. A voltmeter with input resistance of ##1000\Omega## measures the voltage of a 1.5V worn out battery as 0.9V. What is the internal resistance of the battery. Homework Equations ##V=IR## The...
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    Calculating Center of Mass in Cylindrical Coordinates

    Homework Statement Determine the center of mass in cylindrical coordinates of a cone with constant density ##\rho(\vec{r})##. (The cone is inverted, i.e. it's thinnest point is at ##z=0##.) Homework Equations ##m=\int\int\int_C \rho r \, drdzd\theta## ##\overline{r}=\int\int\int_C r\cdot r\...
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    Electron in a Finite Square Well

    Homework Statement An electron in a finite square well has 6 distinct energy levels. If the finite square well is 10nm long determine: a) Approximate the possible values for the depth of the finite square well ##V_0##. b) Using a well depth value in the middle of the results obtained from part...
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    If ##V(x)## is an even function then the energy eigenfunc...

    Homework Statement If ##V(x)## is an even function [i.e. ##V(-x)=V(x)##], then the energy eigenfunctions ##\phi_E(x)## can always be taken to be either even or odd. i.e. show ##\psi_{odd}(x)\equiv\frac{\phi_E(x)-\phi_E(-x)}{2}## and ##\psi_{even}(x)\equiv\frac{\phi_E(x)+\phi_E(-x)}{2}##. The...
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    Rope falling through a hole on a frictionless table

    Homework Statement A rope with uniform density ##\lambda=\frac{m}{L}## is placed on a frictionless table with an initial length ##y_0## hanging through the hole. Derive a differential equation for the position of the bottom of the rope and then using this solve for the time required for the...
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    Number of electrons making it through an NPN transistor

    Homework Statement Suppose ##n## electrons attempt to move through an ##NPN## transistor, there's a probability that some of the electrons traversing the ##P## area will recombine and and not make it to the other side. The infinitesimal probability in a region dx is given by...
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    Complex Impedance of a voltage source and 2 resistors

    Homework Statement A real voltage source can be expressed as an ideal voltage source that is in series with a resistor that represents the inner resistance of the voltage source. This voltage source is a EMF and it is also in series with another resistor. Suppose both the EMF resistor...
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    Finding values for an RLC circuit graphically

    Homework Statement Determine ##q(t=0)##, ##i(t=0)##, the phase difference ##\phi##, angular frequency ##w## and the time constant ##\tau## from the graph of the capacitor waveform below:(pulse voltage source) For this circuit we have ##q(t)=Ae^{-\frac{t}{2\tau}}\cos(\omega t+\phi)##...
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    Potential Difference Across Diode & Resistor

    Homework Statement Assume an ideal diode with ##V_t=0.6##v, find the potential differences across the diode V_ab and across the resistor V_bc as the forward bias voltage is varied from 0 to 10v. Hint: equivalent circuits may be useful Embedding the image wasn't working so...
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