Register to reply

Einstein summation notation for magnetic dipole field

by mmpstudent
Tags: dipole, einstein, field, magnetic, notation, summation
Share this thread:
mmpstudent
#1
Apr30-13, 05:32 PM
P: 16
I can do this derivation the old fashioned way, but am having trouble doing it with einstein summation notation.

Since [itex]\vec{B}=\nabla \times \vec{a}[/itex]
[itex]\vec{B}=\mu_{0}/4\pi (\nabla \times (m \times r)r^{-3}))[/itex]
[itex]4\pi \vec{B}/\mu_{0}=\epsilon_{ijk} \nabla_{j}(\epsilon_{klm} m_{l} r_{m} r^{-3})[/itex]
[itex]=(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})\partial_{j}m_{l}r_{m}r^{-3}[/itex]

here is where I am stumbling. My professor has for the next step

[itex]=m_{l}(\delta_{il}\delta_{jm}-\delta_{im}\delta_{jl})r^{-3} \delta_{jm}-3 r_{m}\hat{r}_{j}r^{-4})[/itex]

but I don't really know how to get to that step
Phys.Org News Partner Physics news on Phys.org
'Squid skin' metamaterials project yields vivid color display
Team finds elusive quantum transformations near absolute zero
Scientists control surface tension to manipulate liquid metals (w/ Video)
Fredrik
#2
Apr30-13, 05:37 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,522
You may be interested in the The LaTeX guide for the forum. Link.
mmpstudent
#3
Apr30-13, 05:43 PM
P: 16
Quote Quote by Fredrik View Post
You may be interested in the The LaTeX guide for the forum. Link.
You were too fast. Was trying to get it to work just needed to delete the spaces in brackets I guess.

Fredrik
#4
Apr30-13, 05:52 PM
Emeritus
Sci Advisor
PF Gold
Fredrik's Avatar
P: 9,522
Einstein summation notation for magnetic dipole field

My first thought is that he's using the product rule for derivatives to evaluate ##\partial_j## acting on a product.
WannabeNewton
#5
Apr30-13, 06:20 PM
C. Spirit
Sci Advisor
Thanks
WannabeNewton's Avatar
P: 5,661
First off, tell your professor that he is horribly butchering Einstein notation. Seriously, what was written down misses the entire point of the notation. Anyways, ##\frac{4\pi}{\mu_{0}}B^{i} = \frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = \epsilon^{ijk}\epsilon_{klm}m^{l}[r^{-3}\partial_{j}r^{m} + r^{m}\partial_{j}(r^{-3})]## hence ##\frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = 2\delta^{[i}_{l}\delta^{j]}_{m}m^{l}[r^{-3}\partial_{j}r^{m} + r^{m}\partial_{j}(r^{-3})]##. Now, ##\partial_{j}r^{m} = \delta^{m}_{j}## and ##\partial_{j}(r^{-3}) = -3(-r^i r_{i})^{-5/2}r_{k}\partial_{j}r^{k} = -3r^{-4}\hat{r}_{j}## giving us ##\frac{4\pi}{\mu_{0}}\epsilon^{ijk}\partial_{j}A_{k} = 2\delta^{[i}_{l}\delta^{j]}_{m}m^{l}[r^{-3}\delta^{m}_{j} -3r^{-4}\hat{r}_{j}r^{m}]## as desired.

EDIT: By the way, in the above it should be ##(r^i r_{i})^{-5/2}## not ##(-r^i r_{i})^{-5/2}##; I've gotten too used to General Relativity xD.
mmpstudent
#6
Apr30-13, 06:35 PM
P: 16
O wow, thanks. that makes much more sense now.
mmpstudent
#7
Apr30-13, 06:55 PM
P: 16
Do you know of any materials online that would give more written out examples of such derivations with Einstein summation? I just need more practice
WannabeNewton
#8
Apr30-13, 07:01 PM
C. Spirit
Sci Advisor
Thanks
WannabeNewton's Avatar
P: 5,661
Quote Quote by mmpstudent View Post
Do you know of any materials online that would give more written out examples of such derivations with Einstein summation? I just need more practice
I honestly can't think of any online resources off of the top of my head because I got used to the notation when learning special relativity (the text used was Schutz).


Register to reply

Related Discussions
Commutative Operators Using Einstein's Summation Notation Advanced Physics Homework 1
Determining the commutation relation of operators - Einstein summation notation Advanced Physics Homework 4
Magnetic field at a dipole (how many atoms surround the dipole?) Introductory Physics Homework 0
Einstein-de Haas Experiment: Magnetic Dipole Moment Alignment Atomic, Solid State, Comp. Physics 2
EINSTEIN Summation notation Differential Geometry 3