- #1
Blanchdog
- 57
- 22
- Homework Statement
- Consider an electric field E = E0 cos(k⋅r - ωt +φ), where k is orthogonal to E0, r is a position vector, and φ is a constant phase. Show that B = (k X E0)/ω * cos(k⋅r - ωt +φ) according to Faraday's Law.
- Relevant Equations
- Faraday's Law: ∇ X E = -∂B/∂t
I got stuck near the beginning, so I tried working backwards. Starting from
B = (k X E0)/ω * cos(k⋅r - ωt +φ)
I found
-∂B/∂t = -k X E0 sin(k⋅r - ωt +φ)
So now I need to find ∇ X (E0 cos(k⋅r - ωt +φ)) and see that it is equal to the above result. This is where I'm stuck though, I'm not sure how to take the curl of this electric field because of that dot product of k and r, leaving the field as a scalar (as far as I can tell). Help is much appreciated!
B = (k X E0)/ω * cos(k⋅r - ωt +φ)
I found
-∂B/∂t = -k X E0 sin(k⋅r - ωt +φ)
So now I need to find ∇ X (E0 cos(k⋅r - ωt +φ)) and see that it is equal to the above result. This is where I'm stuck though, I'm not sure how to take the curl of this electric field because of that dot product of k and r, leaving the field as a scalar (as far as I can tell). Help is much appreciated!