Consistency of Faraday's Law and Electric Field Equation in Optics

[Blue Kangaroo]

Homework Statement

Suppose that an electric field is given by E(r,t)=E₀cos(k·r−ωt+φ), where k⊥E₀ and φ is a constant phase. Show that B(r,t)=((k×E)/ω)B(k⋅r-ωt+φ) is consistent with ∇×E=-∂B/∂t

Homework Equations

∇×E=-∂B/∂t

The Attempt at a Solution

I know I have to take the curl of E, but I'm not sure how to go about doing it.
∂E/∂t would be ωE₀sin(k⋅r-ωt+φ) and ∂E/∂r would be -kE₀sin(k⋅r-ωt+φ), but I'm not sure if that helps. Also, by my calculation, -∂B/∂t= (k×E)cos(k⋅r-ωt+φ).
Any help would be greatly appreciated.

 

[Charles Link]

Suggestion is to write the $E$ field as $E=\vec{ E}_o e^{i (\vec{k} \cdot \vec{x}-\omega t +\phi)}$ and take the real part afterwards. Also, there is a vector calculus identity that $\nabla \times (\psi \vec{a})=(\nabla \psi) \times \vec{a} +\psi (\nabla \times \vec{a})$. The operation then becomes doing a gradient on $e^{i(\vec{k} \cdot \vec{x}-\omega t + \phi)}$. $\\$ And I think you have a typo: the second phase factor $\phi$ should have a plus sign, like the first one. $\\$ Addition note: $\vec{k} \cdot \vec{x}=k_x x+k_y y+k_z z$. You can readily compute the gradient in these Cartesian coordinates. $\\$ Additional suggestion: Let $B=B_o e^{i (\vec{k} \cdot \vec{x}- \omega t+\phi)}$. And what you have should read $B(r,t)=\frac{\vec{k} \times \vec{E}}{\omega} cos(\vec{k} \cdot \vec{r}-\omega t+\phi )$. The rest is straightforward.

 

[Blue Kangaroo]

Suggestion is to write the $E$ field as $E=\vec{ E}_o e^{i (\vec{k} \cdot \vec{x}-\omega t +\phi)}$ and take the real part afterwards. Also, there is a vector calculus identity that $\nabla \times (\psi \vec{a})=(\nabla \psi) \times \vec{a} +\psi (\nabla \times \vec{a})$. The operation then becomes doing a gradient on $e^{i(\vec{k} \cdot \vec{x}-\omega t )}$. $\\$ And I think you have a typo: the second phase factor $\phi$ should have a plus sign, like the first one. $\\$ Addition note: $\vec{k} \cdot \vec{x}=k_x x+k_y y+k_z z$. You can readily compute the gradient in these Cartesian coordinates. $\\$ Additional suggestion: Let $B=B_o e^{i (\vec{k} \cdot \vec{x}- \omega t+\phi)}$. And what you have should read $B(r,t)=\frac{\vec{k} \times \vec{E}}{\omega} cos(\vec{k} \cdot \vec{r}-\omega t+\phi )$. The rest is straightforward.

Thank you, I should be able to take it from here.