Question

Give your answers to each questions clearly. 1) (6 Pts) Write down the Maxwell's equations. Clearly explain the physical meaning of each equation. Consider that both magnetic and electric fields are time-dependent. Give explanation to each term in the equations. 2) (a) Write down the wave equation explicitly in three dimensional form. (4 Pts) (b) Show that the given wave function satisfies the wave equation. (5 Pts) $\psi(x, t) = \frac{1}{a\sqrt{(\pi t)}}e^{-\frac{x^2}{a^2t}}$ (1) (c) Show that the electromagnetic wave of the form satisfies the wave equation for the electric field in spherical coordinates. Hint: Use the Laplace operator given in spherical coordinates. (10 Pts) $\vec{E}(\vec{r}, t) = \vec{E_0}cos(\vec{k}.\vec{r} - \omega t + \phi)$ (2)

          Give your answers to each questions clearly.
1) (6 Pts)
Write down the Maxwell's equations. Clearly explain the physical meaning of each equation.
Consider that both magnetic and electric fields are time-dependent. Give explanation to
each term in the equations.
2)
(a) Write down the wave equation explicitly in three dimensional form. (4 Pts)
(b) Show that the given wave function satisfies the wave equation. (5 Pts)
$\psi(x, t) = \frac{1}{a\sqrt{(\pi t)}}e^{-\frac{x^2}{a^2t}}$
(1)
(c) Show that the electromagnetic wave of the form satisfies the wave equation for the
electric field in spherical coordinates. Hint: Use the Laplace operator given in spherical
coordinates. (10 Pts)
$\vec{E}(\vec{r}, t) = \vec{E_0}cos(\vec{k}.\vec{r} - \omega t + \phi)$
(2)

Give your answers to each questions clearly.
1) (6 Pts)
Write down the Maxwell's equations. Clearly explain the physical meaning of each equation.
Consider that both magnetic and electric fields are time-dependent. Give explanation to
each term in the equations.
2)
(a) Write down the wave equation explicitly in three dimensional form. (4 Pts)
(b) Show that the given wave function satisfies the wave equation. (5 Pts)
ψ(x, t) = (1)/(a√((π t)))e^-(x^2)/(a^2t)
(1)
(c) Show that the electromagnetic wave of the form satisfies the wave equation for the
electric field in spherical coordinates. Hint: Use the Laplace operator given in spherical
coordinates. (10 Pts)
E⃗(r⃗, t) = E⃗0⃗cos(k⃗.r⃗ - ω t + ϕ)
(2)

Added by Randy G.

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