You could divide by ## -u - \frac{b}{k} v## and also by ##m## and then multiply by ##dm## to get
$$\frac{dv}{-u - \frac{b}{k} v} = \frac{dm}{m}$$
This is solvable, see if it gets you the answer you need--I feel like both terms' logarithmic solutions would cancel but because of the coefficients...
Homework Statement
Consider a square slab of magnetized material with sides ##2a##, and
thickness ##d##, as shown. The magnetization is not uniform: $$M = M_o cos(\frac{2πx}{
a}) \hat{z} $$b) Calculate the total magnetic moment of this object. How would
you guess the magnetic field decreases...
I wasn't sure if the uniqueness theorem applied here. Usually you solve for all but one boundary condition and then do a Fourier sum to find one that satisfies all boundary conditions--is that sum the unique solution?
As for the incorrect constant, I see what you mean: I have ##C_x = k^2##...
I think an easy to see answer (that I worked out explicitly, even using the orthogonality method of finding ##C_n## just to be sure) I got $$V(x,y,z)=V_o \frac{sinh(\frac{- \pi (b+c) x}{bc})}{sinh(\frac{- \pi (b+c) a}{bc})} sin(\frac{\pi y}{b}) sin(\frac{\pi z}{c})$$ which like I said is pretty...
Well, kind of. I already know how to solve for every form of ##X(x), Y(y), Z(z)## but my trouble is discerning how I can get rid of certain combinations of ##V(x,y,z)=XYZ##. There are 27 total combinations, but the constant equation limits to I think 18. I was hoping I could reason, "well, we...
Sorry about the formatting, my post seems unable to be edited now.
Boundaries:All faces $$V=0$$ except for $$V(a,y,z)=V_o sin(\frac{\pi y}{b}) sin(\frac{\pi z}{c})$$ at x=a
Dimensions: ##a*b*c##
Relevant Equations: ##\frac{X''}{X} + \frac{Y''}{Y} + \frac{Z''}{Z }= 0##
implies ##C_x + C_y...
Homework Statement
An #a*b*c box is given in x,y,z (so it's length #a along the x axis, etc.). Every face is kept at #V=0 except for the face at #x=a , which is kept at #V(a,y,z)=V_o*sin(pi*y/b)*sin(pi*z/c). We are to, "solve for all possible configurations of the box's potential"
Homework...
I thought so, so it all worked out--the potential is 0 everywhere outside of the sphere, and I was able to prove the image charge use by plugging in all points on the radius to show the terms cancel, so I believe I've solved it completely. Thank you very much for the help!
Is infinity considered the other bound of the volume outside of the sphere? I'm not sure if it should be zero or finite at infinity--both are satisfied anyways.
For my outside solution, I tried using Laplace's eqn. coupled with: V=0 everywhere on the shell and rho = 0 everywhere outside of the shell. I got a solution of form ##V = -C/r + D## (because there is no angular dependence) and I'm assuming that the -C/r term comes from the point charge, so...
Homework Statement
I have a hollow, grounded, conducting sphere of radius R, inside of which is a point charge q lying distance a from the center, such that a<R. The problem claims, "There are no other charges besides q and what is needed on the sphere to satisfy the boundary condition".
I...