Hi, I just wanted to let you know that there is a brand new Physics Flash animation web site out there. It contains interactive content that can be used in a classroom setting (used by professors or teachers) or by students themselves. The level of the material presented is an introductory level...
Hi, I was curious how one would solve this problem:
In a town, there are 1850 trees along public roads. Each year, the town has to remove on average 25 trees of random age because of various reasons (natural death, fungus infection, insects, hit by cars, roots damaged by construction, etc.)...
I'm trying to calculate the outgassing rate for the following problem, but I seem to be missing something (maybe the like the pumping speed or time).
The base pressure of a particular vacuum system is 10^-9 torr. A sample with a fingerprint of area ~1 cm^2 is introduced into the system...
I have trouble doing a problem involving exact differentials:
Consider a uniform wire of length L and cross-section area A. A force F is applied to the wire. We can write the relationship:
dL = (L/YA)dF + (aL)dT
where Y is the Young's modulus, a the coefficient of thermal expansion, and...
Hello, I have trouble showing that the following initial-value problem has a unique solution. I also need to find this unique solution.
y' = e^(t-y), where 0 <= t <= 1, and y(0) = 1.
How can I test the Lipschitz condition on this?
Thanks in advance.
Hi, could someone show me how to express
\frac{\partial G^{\mu\nu}}{\partial x^\nu} = 0
which are Maxwell's equations, G is the dual tensor,
in terms of the field tensor F:
\frac{\partial F_{\mu\nu}}{\partial x^\lambda} + \frac{\partial F_{\nu\lambda}}{\partial x^\mu} + \frac{\partial...
Hi, I need help on the following problem on Fourier series:
Let phi(x)=1 for 0<x<pi. Expand
1 = \sum\limits_{n = 0}^\infty B_n cos[(n+ \frac{1}{2})x]
a) Find B_n.
b) Let -2pi < x < 2pi. For which such x does this series converge? For each such x, what is the sum of the series?
c) Apply...
Hi, I need help on the following question: Suppose a point charge q is constrained to move along the x-axis. Show that the fields at points on the axis to the RIGHT of the charge are given by
\vec{E} = \frac{q}{4\pi\epsilon_0}\frac{1}{r^2}(\frac{c+v}{c-v})\hat{x} and \vec{B} = 0
What...
Hi, I have a question about Taylor series:
I know that for a function f(x), you can expand it about a point x=a, which is given by:
f(x) = f(a) + f'(a)(x - a) + \frac{f''(a)}{2!}(x - a)^2 + ...
but I would like to do it for f(x+a) instead of f(x), and expand it about the very same point...
I'm stuck on the following problem:
A long thin coil of length l, cross-sectional area S, and n turns per unit length carries a current I. It is placed along the axis of a large circular ring of radius a, which is carrying a current I'. If d is the displacement of the center of the coil from...
This is a numerical analysis question, and I am trying to prove that the p(0), p'(0), p(1), p'(1) define a unique cubic polynomial, p. More precisely, given four real numbers, p00, p01, p10, p11, there is one and only one polynomial, p, of degree at most 3 such that p(0) = p00, p'(0) = p01...
I'm stuck on the following eigenvalue problem:
u^{iv} + \lambda u = 0, 0 < x < \pi
with the boundary conditions u = u'' = 0 at x = 0 and pi.
("iv" means fourth derivative)
I look at the characteristic polynomial for lambda > 0 and < 0 and I get fourth roots for each of them. In the case...
Hi, I have trouble constructing the proof for the existence of a solution u that vanishes at some point in an open interval (a, b) for the Sturm-Liouville differential equation:
\frac{d}{dx} (P(x) \frac{du}{dx}) + Q(x)u = 0
We can assume that P is continuously differentiable and greater...
I have been trying to apply to summer REU (research experience for undergraduates) programs, in particular in material science and nanoscience. Last year I didn't get into any of the four I applied to. This year I applied to another six, and so far I have received one denial. I'm beginning to...
Hi, I'm stuck on this problem:
Two solid hemispherical conducting electrodes each of radius r=a are pressed into the earth, curved surfaces down, such that the flat surfaces are flush with the (flat) earth's surface. Assume that the electrode separation (center to center) is d where d >> a...
Hi, can anyone help me prove the following:
Show that the nonwandering set is closed and positively invariant.
I always have trouble working with sets because they're so abstract. If anyone can help me, that would be great. Thanks.
Definition of nonwandering set is here...
Hi, I'm having trouble doing this problem:
A truncated conical cylinder of graphite (bulk resistivity \rho = 1/\sigma ). The top of the cylinder has radius r = a, the bottom has r = b (b>a). Find the effective resistance between top and bottom of the cylinder. Show that the expression reduces...
Hi, I have trouble on the following problem:
Given a parallel plate capacitor, fixed area A, and fixed separation d. Find the energy stored, before and after insertion of a slab of dielectric, which completely fills the space between plates, for each of the two cases:
a) Plates are...
Hi, I'm having trouble applying Laplace's equation solution in cylindrical coordinates to the problem of the grounded conducting cylinder of radius a in a uniform external field. The cylinder axis is the z axis, and the external electric field is E0 in the x direction. I need to find the...
This is another question I have trouble proving:
Suppose the coefficients of the equation: w'' + p(z)w' + q(z)w = 0 are analytic and single-valued in a punctured neighborhood of the origin. Suppose it is known that the function w(z) = f(z) ln z is a solution, where f is analytic and...
Could someone show me how to find the indicial equation and the indicies relative to any regular singular point of the Legendre equation:
(1 - z^2)w'' - 2zw' + kw = 0
Thank you.
Hi, I would like some help in proving the following:
Consider the action for a particle in a potential U. Show that an extremum path is never that of a local maximum for the action.
I think what I have to do is look at the second derivative of the action integral. Then I should somehow...
Hi, I'm trying to solve a differential equation and I'm supposed to obtain a recursion formula for the coefficients of the power series solution of the following equation:
w'' + (1/(1+z^2)) w = 0.
The term 1/(1+z^2) I recognize as a geometric series and can be expressed as sum of 0 to...
I'm trying to find the dipole moment of a non-uniform surface charge distribution on a sphere of radius a:
The surface charge distribution is:
\sigma = \sigma_{0} cos \theta
where theta is the polar angle.
Here is what I did:
\vec{p} = \int\vec{r}\sigma dA
= \int r \sigma_{0} cos...
Hi, this might be very easy, but I forgot how to do the following: I have a vector in R^6: (x1, x2, x3, x4, x5, x6). How do I find a vector such that their dot product vanishes? I know how to do it for the two dimensional case: (x1, x2), so the vector that is perpendicular to it is c(-x2, x1)...
I need help on how to do this problem: Carry out the calculation on the simplest quadrupole: Two point dipoles are oppositely oriented along the z-axis, separated by distance a. The potential due to one dipole is V = (p cos(theta) / (4pi epsilon r^2). The result I should get for the quadrupole...
Hi, I have no idea of how to do the following problem and what formulas I should use. Please help! Thank you.
A spherical balloon has a conductive coating and we propose to inflate the balloon to a diameter of 0.1 meters by connecting the surface to a high voltage source. Suppose that the...
Please consider this problem:
A plastic slab of thickness t has a uniform free charge density, +rho, embedded inside, and also one surface has a surface charge of -sigma. Find the electric fields here and sketch as a function of distance from one surface. Also find the potential as a function...
Hi, I would like to ask a question about grounding conductors. Suppose we have concentric spherical thin conducting shells (consisting of an inner conductor and an outer conductor). Suppose a charge of +Q is placed on the inner conductor. If we ground the outer conductor, my understanding is...