Additional Problems Start from Eqs. 33-11 and 33-17 and show...

Additional Problems Start from Eqs. 33-11 and 33-17 and show that $E(x, t)$ and $B(x, t),$ the electric and magnetic field components of a plane traveling electromagnetic wave, must satisfy the "wave equations" $$\frac{\partial^{2} E}{\partial t^{2}}=c^{2} \frac{\partial^{2} E}{\partial x^{2}} \quad \text { and } \quad \frac{\partial^{2} B}{\partial t^{2}}=c^{2} \frac{\partial^{2} B}{\partial x^{2}}.$$

Additional Problems
Start from Eqs. 33-11 and 33-17 and show that $E(x, t)$ and $B(x, t),$ the electric and magnetic field components of a plane traveling electromagnetic wave, must satisfy the "wave equations"
$$\frac{\partial^{2} E}{\partial t^{2}}=c^{2} \frac{\partial^{2} E}{\partial x^{2}} \quad \text { and } \quad \frac{\partial^{2} B}{\partial t^{2}}=c^{2} \frac{\partial^{2} B}{\partial x^{2}}.$$

Key Concepts

Maxwell's Equations

These are the fundamental equations governing electric and magnetic fields. They are essential for describing how field variations in space and time generate electromagnetic phenomena and provide the basis for deriving the wave equations for electric and magnetic fields in free space.

Wave Equations

The wave equation is a second order partial differential equation that models how physical quantities, such as the electric and magnetic fields in electromagnetic waves, propagate through space and time at a constant speed. Its form demonstrates the intrinsic relationship between spatial and temporal variations in the fields.

Plane Electromagnetic Waves

These are idealized solutions of Maxwell's equations where the wave fronts (surfaces of constant phase) are infinite, parallel planes. This simplification helps in directly applying mathematical techniques to derive the wave equation, as these waves have uniform properties in planes perpendicular to the direction of propagation.

Partial Differential Equations

PDEs are mathematical tools used to express the relationships involving the rates of change with respect to more than one independent variable, such as space and time. In the context of electromagnetic waves, these equations capture the essential behavior and dynamics (e.g., propagation, dispersion) of the fields.

Transcript

00:01 First part, we'll call it part a.

00:02 We're trying to prove that the second derivative of the electric field with respect to time would be equal to the speed of light squared times the second derivative of the electric field with respect to the position.

00:14 So we can say the partial derivative of e with respect to x would be equal to the negative partial derivative of the magnetic field with respect to t time.

00:26 We can then say we're going to differentiate partially with respect to x one more time.

00:34 We can say the partial derivative, the second partial derivative of the electric field with respect to x would be equal to the partial derivative with respect to x of the partial derivative of the electric field with respect to x.

00:59 We know that here this is simply going to be equal to the partial derivative with respect to x multiplied by negative the partial derivative of the magnetic field with respect to time and then we're going to say that here this would be equal this would be equal to the negative partial derivative with respect to time multiplied by negative times the uh magnetism permeability is magnetic permeability mu not times epsilon not times the partial derivative of the electric field with respect to time and this is going to be equal to mu not epsilon not times the second derivative of the electric field with respect to time and here we can save for part a therefore the second derivative of the the electric field with respect to time would be equal to 1 over mu not epsilon not times d squared rather the second partial derivative of the electric field with respect to x and so substituting in we find that the partial derivative of the electric field with respect to time the second partial derivative of the electric field with respect to time would equal the speed of light squared, multiplied by the second partial derivative with respect to position.

02:55 This would be proving part a...

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