Calculation of magnetic field from electric field


by maggas
Tags: calculation, electric, field, magnetic
maggas
maggas is offline
#1
May18-09, 07:41 AM
P: 2
1. The problem statement, all variables and given/known data
In this tute on EM waves, we were given the Electric Field

[tex] \textbf{E}=\textbf{E}_0\text{exp}(i(\textbf{k}\cdot\textbf{x} - \omega t))[/tex]

which after a fair bit of algebra yields the magnetic field

[tex] \textbf{B}=(\hat{\textbf{k}}\times\textbf{E})/c[/tex]

Similarly the inverse problem I had to solve, given the Magnetic Field

[tex] \textbf{B}=\textbf{B}_0\text{exp}(i(\textbf{k}\cdot\textbf{x} - \omega t))[/tex]

yields [tex] \textbf{E}=c\textbf{B}\times\hat{\textbf{k}}[/tex]

The tute also gives a hint that this can be solved in a few lines, without heavy algebra, using Lagrange's formula.

2. Relevant equations

Maxwell's Equations

Lagrange's formula: [tex]\textbf{a}\times(\textbf{b}\times\textbf{c}) = (\textbf{a}\cdot\textbf{c})\textbf{b} - (\textbf{a}\cdot\textbf{b})\textbf{c}[/tex]

3. The attempt at a solution

I can only solve the question the long winded way, and would like to know how it can be solved using this identity rather than equating many equations to solve coefficients!

EDIT: Forgot to mention only using simplified Maxwell's Equations, i.e. Gauss' = 0 and Ampere's has no J term
Phys.Org News Partner Science news on Phys.org
NASA's space station Robonaut finally getting legs
Free the seed: OSSI nurtures growing plants without patent barriers
Going nuts? Turkey looks to pistachios to heat new eco-city
maggas
maggas is offline
#2
May19-09, 03:46 PM
P: 2
I've managed to solve this myself, thanks for the help

[tex]\textbf{B}=(\hat{\textbf{k}}\times\textbf{E})/c[/tex]
[tex]c\textbf{B}=(\hat{\textbf{k}}\times\textbf{E})[/tex]

Then using Lagrange Triple Product
[tex]\hat{\textbf{k}}\times(\hat{\textbf{k}}\times\textbf{E}) = (\hat{\textbf{k}}\cdot\textbf{E})\hat{\textbf{k}} - (\hat{\textbf{k}}\cdot\hat{\textbf{k}})\textbf{E}[/tex]
[tex]\hat{\textbf{k}}\times c\textbf{B} = - \textbf{E}[/tex]

Therefore
[tex]\textbf{E} = c\textbf{B} \times \hat{\textbf{k}}[/tex]
//as required


Register to reply

Related Discussions
angular momentum in electric field and magnetic field Classical Physics 5
Electromagnetic Waves Question, regarding Magnetic Field and Electric Field Introductory Physics Homework 25
electric field and magnetic field - proton deflection Introductory Physics Homework 6
Magnetic Field and Electric field outside a long solenoid Advanced Physics Homework 5
Electric Field/Magnetic Field Questions Introductory Physics Homework 1