Question

In Section 4.3, we claim that in analyzing electromagnetic waves, we could handle the fieldsandtogether with complex numbers. Show that if we define an "electromagnetic field"$\mathbf{\text{G}}\mathbf{\equiv }\mathbf{\text{E+icB}}$, then the two of Maxwell's equations that link$\mathit{E}$and$\mathit{B}$.$\mathbf{\text{(4-6c)}}$ and$\mathbf{\text{(4-6d)}}$ , become just one:

$\mathbf{\oint }\mathbf{\text{G}}\mathbf{\cdot }\mathbf{\text{dl}}\mathbf{=}\frac{\mathbf{\text{i}}}{\mathbf{\text{c}}}\frac{\mathbf{\partial }}{\mathbf{\partial }\mathbf{\text{t}}}\int \text{G}\mathbf{\cdot }\mathbf{\text{dA}}$

Electromagnetic waves would have to obey this complex equation. Does this change of approach make $\mathit{E}$and$\mathit{B}$/or complex? (Remember how a complex number is defined.)

Short answer

Start with Maxwell's equations and utilize the identity $\text{G=E+icB}$to get to the given equation; the electric and magnetic fields don't have to be complicated.

Step-by-step solution

Significance of Maxwell’s equations

Maxwell's equations formulate a relationship between the electric and magnetic fields, that how they are correlated and complementary to each other.

Step 2:Maxwell’s equation to relate  Eand  B.

We should be able to merge equations $\text{(4-6c)}$and $\text{(4-6d)}$into a single equation for both $E$and $B$ if we start with equations $\text{(4-6c)}$and $\text{(4-6d)}$and use the supplied identity .

$\text{G=E+icB}$

$\oint \text{E}\cdot \text{dl}=-\frac{\partial }{\partial t}\int \text{B}\cdot \text{dA}$……………….. (1)

$\oint \text{B}\cdot \text{dl}=\frac{\text{1}}{{\text{c}}^{\text{2}}}\frac{\partial }{\partial t}\int \text{E}\cdot \text{dA}$……………….. (2)

Equation(2) is multiplied by$\text{ic}$, and equations (1) and (2) are added together.

$\oint \text{E}\cdot \text{dl}+\text{ic}\oint \text{B}\cdot \text{dl}=-\frac{\partial }{\partial t}\int \text{B}\cdot \text{dA}+\frac{\text{i}}{\text{c}}\frac{\partial }{\partial t}\int \text{E}\cdot \text{dA}$

$\oint \left(\text{E+icB}\right)\cdot \text{dl}=\frac{\text{i}}{\text{c}}\frac{\partial }{\partial t}\int \left(\text{icB+E}\right)\cdot \text{dA}$

Since, $\text{G=E+icB}$

$\oint \text{G}\cdot \text{dl}=\frac{\text{i}}{\text{c}}\frac{\partial }{\partial t}\int \text{G}\cdot \text{dA}$

As a result, we were able to unify the two equations for electric and magnetic fields into a single equation that includes both in a single variable termed as $G$. The magnetic and electric fields, on the other hand, haven't been restricted in any way during the derivation, thus they don't have to be complicated.