# About the formation of an electromagnetic wave

I was going through the derivation for the equation electromagnetic waves and got these equations $$\mu\epsilon \frac{\partial^2\mathbf{E}}{\partial t^2} = \mathbf{\nabla}\times\left(\frac{\partial\mathbf{B}}{\partial t}\right) = - \mathbf{\nabla}\times(\mathbf{\nabla}\times\mathbf{E}) = \mathbf{\nabla}^2\mathbf{E}$$

Then, I tried to physically visualize this by imagining this situation. For $$\frac{\partial^2\mathbf{E}}{\partial t^2}$$ to exist, there needs to be an acceleration of a charge. Once the charge accelerates, there is a change in the magnetic field $$\mathbf{\nabla}\times\left(\frac{\partial\mathbf{B}}{\partial t}\right)$$ (if the charge were not accelerating, there would just be a constant magnetic field). Since the magnetic field changes, it produces an electric field in the opposite direction (To oppose the change in the flux of the magnetic field). This again causes a change in the magnetic field and the electric field is again increases along the opposite direction.

Is it right to think of it that way, to get an electromangetic wave?

## 1 Answer

What you need to understand is the so-called but erroneously called "Jefimenko's Equations" [2] but the way they were written by McDonald [1], see there his equations (4) and (5) in Gaussian units and repeated here in MKS-SI units (similar equations eqs. 14-33 and 14-36 are developed in [3]):

$$\mathbf{B} = \frac{\mu_0}{4\pi}\int \frac{ [\mathbf J] \times \hat {\mathbf{n}} }{R^2} d\mathbf{x}' + \frac{\mu_0}{4\pi}\frac{1}{c}\int \frac{ ([\mathbf {\dot J}] \times \hat {\mathbf{n}})}{R} d\mathbf{x}'\\ \mathbf{E} = \frac{1}{4\pi\epsilon_0} \int \frac{[\rho]\hat {\mathbf{n}}}{R^2}d\mathbf{x}' + \frac{1}{4\pi\epsilon_0} \frac{1}{c}\int \frac{ ([\mathbf J] \cdot \hat {\mathbf{n}}) \hat {\mathbf{n}} + ([\mathbf J] \times \hat {\mathbf{n}})\times \hat {\mathbf{n}} }{R^2} d\mathbf{x}'\\ +\frac{1}{4\pi\epsilon_0} \frac{1}{c^2}\int \frac{ ([\mathbf {\dot J}] \times \hat {\mathbf{n}})\times \hat {\mathbf{n}} }{R} d\mathbf{x}'$$

The brackets $$[]$$ signify retarded times, $$\mathbf R =\mathbf x -\mathbf x'$$, $$\hat{\mathbf{n}}=\mathbf{R}/R$$ and $$\mathbf{\dot J}=\partial \mathbf{J}/\partial t$$. These equations are equivalent to the Maxwell's equations in homogeneous matter. The last term in both the B and E equations depend as $$1/R$$ showing explicitly the radiation induced by the time-varying current $$\mathbf{\dot J}$$ in the far field, the so-called radiation terms.

Note how the transversality of the radiated wave is shown explicitly by $$\mathbf B \sim [\mathbf {\dot J}] \times \hat {\mathbf{n}}\\ \mathbf E \sim ([\mathbf {\dot J}] \times \hat {\mathbf{n}})\times \hat {\mathbf{n}}$$ that is $$\mathbf B , \mathbf E , \hat {\mathbf n}$$ are orthogonal to each other and to the direction of propagation in the far-field.

[1] McDonald: "The relation between expressions for time-dependent electromagnetic fields given by Jefimenko and by Panofsky and Phillips," Am. J. Phys. 65 (11), November 1997

[3] Panofsky, Phillips: Classical Electricity and Magnetism, 2nd edition, 1962