# Search results

1. ### Derivation: force on magnetic moment in magnetic field

I am trying to derive the equation ##\vec{\Gamma} = \vec{m} \times \vec{B}##, where ##\vec{m} = I \vec{A}## is the magnetic moment, and ##\vec{A}## is normal to surface ##A##, from the Lorentz force law ##\mathrm{d}\vec{F} = I \vec{dl} \times \vec{B}##. For an arbitrary closed current loop ##C##...
2. ### A Asymptotic solutions to a differential equation

I am asked to solve the differential equation $$f''(\eta)+\frac{f'(\eta)}{\eta}+\Big(1-\frac{s^2}{\eta^2}\Big) f(\eta) - f(\eta)^3 = 0,$$ for small ##\eta## and large ##\eta## under the condition ##f(\eta \rightarrow \infty) = 1## and ##f(0)=0##. The numerically solved solution looks like...
3. ### Diffusion equation in polar coordinates

Homework Statement I am trying to solve the axisymmetric diffusion equation for vorticity by Fourier transformation. Homework Equations $$\frac{\partial \omega}{\partial t} = \nu \Big( \frac{1}{r}\frac{\partial \omega}{\partial r} + \frac{\partial^2 \omega}{\partial r^2} \Big).$$ The...
4. ### Taylor expansion

Hello friends, I need to compute the taylor expansion of $$\frac{x^4 e^x}{(e^x-1)^2},$$ for ##x<<1##, to find $$x^2 + \frac{x^4}{12}.$$ Can someone explain this to me? Thanks!