From Gauss's Law
give ##E=\dfrac{\sigma}{2\epsilon_0}##
##\therefore P_e=\dfrac{\sigma^2}{2\epsilon_0}##
Consider at equilibrium (before bubble being charged)
##P_i=P_0+\dfrac{4S}{R}##
Using Newton's 2nd Law
##\Sigma F=m\ddot{R}##
Let ##R+\delta R## be the new radius
Give (after binomial...
I was initially curious by the fact that streamlines around a circle appear the same as the lines of stress around a hole:
I understand that streamlines are the contour lines of the stream function ##\psi## which satisfies the Laplace equation. I was wondering there is a related function for...
Assume that an infinite metallic plate A lies in the xy-plane, and another infinite metallic plate B is parallel to A and at height z = h.
The potential of plate A is 0, and the potential of plate B is constant and equal to V.
So, there is a uniform electrostatic field E between plates A and B...
After looking around a bit, I found that, considering the polar axis to be along the direction of the point charge as suggested by the exercise, the following Legendre polynomial expansion is true:
$$\begin{equation}\frac{1}{|\mathbf{r} - \mathbf{r'}|} = \sum_{n=0}^\infty...
Homework Statement
There's a metal cunducting cube with edge length ##a##. Three of its walls: ##x=y=z=0## are grounded and the other three walls: ##x=y=z=a## are held at a constant potential ##\phi_{0}## . Find potential inside the cube.
Homework Equations
The potential must satisfy Laplace...
The time-dependent Schrodinger equation is given by:
##-\frac{\hslash^{2}}{2m}\triangledown^{2}\psi+V\psi=i\hslash\frac{\partial }{\partial t}\psi##
Obviously, there is a laplacian in the kinetic energy operator. So, I was wondering if the equation was rearranged as...
I am solving the Laplace equation in 3D:
\nabla^{2}V=0
I am considering azumuthal symmetry, so using the usual co-ordinates V=V(r,\theta). Now suppose I have two boundary conditions for [V, which are:
V(R(t)+\varepsilon f(t,\theta),\theta)=1,\quad V\rightarrow 0\quad\textrm{as}\quad...
Homework Statement
An Ohmic material with some conductivity has a uniform current density J initially. Let's say the current is flowing in the direction of the z-axis. A small insulating sphere with radius R is brought inside the material. Find the potential outside the sphere.
Homework...
Homework Statement
The problem states:
"A point charge q is located at a fixed point P on the internal angle bisector of a 120 degree dihedral angle between two grounded conducting planes. Find the electric potential along the bisector."
Homework Equations
ΔV = 0
with Dirichlet boundary...
Homework Statement
Two concentric cylinders with radii a & b (b>a) with an infinitely long grounded strip along the z-direction are given potentials \phi_1 and \phi_2.
Find \Phi(r,\phi) for a<r<b
Boundary conditions:
\Phi(r,2n\pi)=0
\Phi(a,\phi)=\phi_1
\Phi(b,\phi)=\phi_2
Homework...
Homework Statement
House: a room (see figure) has perfectly isolated walls, except the two windows
where a convective heat exchange takes place (with the same transfer coefficient).
Outside temperature in front of a sun-faced wall-sized panoramic window is T1,
while at the back it is...
I am studying the linear oscillation of the spherical droplet of water with azimuthal symmetry. I have written the surface of the droplet as F=r-R-f(t,\theta)\equiv 0.
I have boiled the problem down to a Laplace equation for the perturbed pressure, p_{1}(t,r,\theta). I have also reasoned that...
Q) A conducting sphere of radius R floats half submerged in a liquid dielectric medium of permittivity e1. The region above the liquid is a gas of permittivity e2. The total free charge on the sphere is Q. Find a radial inverse-square electric field satisfying all boundary conditions and...
Consider an inﬁnitely long hollow dielectric cylinder of radius a with the electricpotential V = V0 cos φ on the surface of the cylinder where φ is an angle measured around the axis of the cylinder. Solve Laplace’s equation to ﬁnd the electric potential everywhere in space.
Do you just plug V...
Suppose u(x,y) and v(x,y) are harmonic on G and c is an element of R. Prove u(x,y) + cv(x,y) is also harmonic.
I tried using the Laplace Equation of Uxx+Uyy=0
I have:
du/dx=Ux
d^2u/dx^2=Uxx
du/dy=Uy
d^2u/dy^2=Uyy
dv/dx=cVx
d^2v/dx^2=cVxx
dv/dy=cVy
d^2v/dy^2=cVyy
I'm not really sure how to...
Hi! Everyone. I encounter some trouble in deriving the kernel of Laplace equation with prescribed boundary conditions.
Given the following preposition:
$$T(x, y) = \int_{-\infty}^{\infty}dx'\frac{y/\pi}{(x-x')^{2}+y^2}F(x')......[1]$$
satisfies the Laplace equation for ##x\in(-\infty, \infty)##...
Homework Statement
A ring charge of total charge Q and radius a is concentric with a grounded conducting sphere of radius b, b < a.
Determine the potential everywhere. The ring is located in the equatorial plane, so both the sphere and the ring have their center at the same spot.
Homework...
Homework Statement
Homework Equations
Heat equation
The Attempt at a Solution
I can derive E(t) to get integral of du/dt over 0 to L, which is the same as integrating the right hand side of the original equation (d2u/dx2+sin(5t); while this allows me to take care of the d2u/dx2, I don't know...
Homework Statement
Consider an infinite sheet of magnetized tape in the x-z plane with a nonuniform periodic magnetization M = cos(2πx/λ), where λ/2 is the distance between the north and south poles of the magnetization along the x-axis. The region outside the tape is a vacuum with no currents...
Homework Statement
I'm having issues with a Laplace problem. actually, I have two different boundary problems which I dont know how to solve analytically.
I couldn't find anything on this situations and if anybody could point me in the right direction it would be fantastic.
It's just Laplace's...
Homework Statement
Homework Equations
Is my part a correct and am I on the right track for part b? If not please give me some suggestions to get me closer to the right track. Also, how would I even begin c.? We have literally done no examples like this in class.
The Attempt at a Solution...
I learn that we can expand the electric potential in an infinite series of rho and cos(n*phi) when solving the Laplace equation in polar coordinates. The problem I want to consider is the expansion for the potential due to a 2D line dipole (two infinitely-long line charge separated by a small...
So It is well known that the 2D solution to the Laplace equation can be obtained by changing to complex coordinates ##u=x+iy## and ##v=x-iy##. This can be extended to n dimensions as long as the complex coordinates chosen also solve the Laplace equation. For example in 3D...
I'm stuck on a seemingly simple 2D electrostatics problem. The problem is as follows:
A parabolic interface ($$x(y)=cy^2$$) separates two regions of different conductivities, with a uniform electric field at infinity aligned with the x-axis.
I write the Laplace operator in parabolic...
The angular equation:
##\frac{d}{d\theta}(\sin\theta\,\frac{d\Theta}{d\theta})=-l(l+1)\sin\theta\,\Theta##
Right now, ##l## can be any number.
The solutions are Legendre polynomials in the variable ##\cos\theta##:
##\Theta(\theta)=P_l(\cos\theta)##, where ##l## is a non-negative integer...
1. The problem statement, all variables a
nd given/known data
A rectangular trough extends infinitely along the z direction, and has a cross section as shown in the figure. All the faces are grounded, except for the top one, which is held at a potential V(x) = V_0 sin(7pix/b). Find the...
< Mentor Note -- thread moved to HH from the technical math forums, so no HH Template is shown >
So this is the problem
Here is the question:
A 32lb weight strecthes a spring 2ft.The weight is released from rest at the equilibrium position. beginning at t=0, a force equal to f(t)= sint acts on...