1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Maxwell's Equations in Vacuum: Constraints on Wave

  1. Feb 5, 2013 #1
    1. The problem statement, all variables and given/known data

    Condensed/simplified problem statement

    [itex]\vec{E} = f_{y}(x-ct)\hat{y} + f_{z}(x-ct)\hat{z} \\
    \vec{B} = g_{y}(x-ct)\hat{y} + g_{z}(x-ct)\hat{z} \\[/itex]

    All the f and g functions go to zero as their parameters go to ±∞.

    Show that gy = fz and gz = -fy

    2. Relevant equations
    [itex]\nabla \cdot \vec{E} = 0 \\
    \nabla \cdot \vec{B} = 0 [/itex]
    [itex]\nabla \times \vec{E} = -\frac{\partial \vec{B}}{\partial t} [/itex]
    [itex]\nabla \times \vec{B} = \frac{1}{c^{2}}\frac{\partial \vec{E}}{\partial t}

    3. The attempt at a solution
    [itex]\nabla \times \vec{E} = -\frac{\partial f_{z}(x-ct)}{\partial x}\hat{y} + \frac{\partial f_{y}(x-ct)}{\partial x}\hat{z} = -\frac{\partial \vec{B}}{\partial t} = \\-(g^{'}_{y}(x-ct)(-c)\hat{y} + g^{'}_{z}(x-ct)(-c)\hat{z}) = c(g^{'}_{y}(x-ct)\hat{y} + g^{'}_{z}(x-ct)\hat{z}) [/itex]

    The -c factors come from the chain rule when differentiating B with respect to t. So then I have the following.

    [itex]-\frac{\partial f_{z}(x-ct)}{\partial x} = -f^{'}_{z}(x-ct) = cg^{'}_{y}(x-ct) \\
    \frac{\partial f_{y}(x-ct)}{\partial x} = f^{'}_{y}(x-ct) = cg^{'}_{z}(x-ct)[/itex]

    The problem is to relate the functions directly and not by their derivatives. We are given that the functions all go to zero as the parameter goes to +/- infinity. So maybe it would be okay to just integrate without concern for a constant added on. But supposedly the c factor isn't there. The answer is supposed to have[itex]g_{y} = -f_{z}[/itex]. How do I get there?
    Last edited: Feb 6, 2013
  2. jcsd
  3. Feb 6, 2013 #2


    User Avatar
    Science Advisor
    Homework Helper
    Gold Member

    B and E do not have the same units, so f and g must be related with an appropriate factor of c. There's probably a typo in the problem. Your argument about the integration constants makes sense.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook