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) =...
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}##...
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...
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...
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...
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...
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...
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
# # # # # # #...
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
|...
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...
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...
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$$
Homework Equations
The Attempt at a Solution
##u=2\sqrt{x}##...
Homework Statement
Given two indistinguishable objects at the same initial temperature ##T_i##, calculate the minimum work done by a refrigerator functioning between the two objects till one of the objects reaches a new temperature ##T_2##, assume constant heat capacities and constant pressure...
Homework Statement
The temperature across the capillary with constant cross-sectional area and length L is given by ##T=T_0e^{-kx}##. Assuming an ideal gas and constant pressure show the number of moles to be: $$n=\frac{PV(e^{kL} - 1)}{RkLT_0}$$
Homework Equations
##PV=nRT##
The Attempt at a...
Homework Statement
n moles of an ideal gas are placed in a frictionless piston with weight ##w_p## and cross-sectional area ##A##. The quantity ##\gamma = \frac{c_p}{c_v}## is a constant, the gas is originally at equilibrium values##(P_i, V_i, \theta_i)## and the external pressure is taken to...
Homework Statement
Suppose a rain drop with mass ##m_0\neq 0## is falling due to gravity with initial velocity ##v_0##, assume ##\frac{dm}{dt}=k=##constant. Solve the differential equation and determine the velocity as ##t\to\infty##
Homework Equations
##F=\frac{dp}{dt}=\dot{m}v+m\dot{v}##
The...
Homework Statement
The equation for the normalized ##n=3##, ##l=2##, ##m=0## wavefunction is given by $$\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>##...
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...
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...
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...
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...
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\...
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...
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...
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...
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...
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...
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)##...
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...