Question

${\mathit{e}}^{\mathbf{\gamma }}$,where $\mathit{\gamma }\mathbf{=}\mathit{a}\mathbf{+}\mathit{j}\mathit{\beta }$, can be expressed as ${\mathit{e}}^{\mathbf{\alpha }\mathbf{I}}\mathbf{\angle }\mathit{\beta }\mathit{I}$, in which $\mathit{\alpha }\mathit{I}\mathit{i}\mathit{s}$is dimensionless and $\mathit{\beta }\mathit{I}\mathit{i}\mathit{s}$is in radians (also dimensionless).

1. True
2. False

Short answer

Therefore, the correct option is (a) True.

Step-by-step solution

Determine the formula of propagation constant.

Write the formula of propagation constant.

$\mathit{\gamma }\mathbf{=}\mathbf{(}\mathbf{\alpha }\mathbf{+}\mathbf{j}\mathbf{\beta }\mathbf{)}{\mathit{m}}^{\mathbf{-}\mathbf{1}}$ ……. (1)

Here, $\gamma$ is propagation constant.

Determine the correct option.

The given expression is,

${\mathit{e}}^{\mathbf{\gamma }\mathbf{\iota }}$ ….. (2)

Substitute$\mathit{\alpha }\mathbf{+}\mathit{j}\mathit{\beta }$for $\mathit{\gamma }$ in expression (2).

$$\begin{gathered} {\mathit{e}}^{\mathbf{\gamma }\mathbf{\iota }}\mathbf{=}{\mathit{e}}^{\left(\alpha +j\beta \right)\mathbf{\iota }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathit{e}}^{\mathbf{\alpha }\mathbf{\iota }}{\mathit{e}}^{\mathbf{j}\mathbf{\beta }\mathbf{\iota }} \\ \mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{}\mathbf{=}{\mathit{e}}^{\mathbf{\alpha }\mathbf{\iota }}\mathbf{\angle }\mathit{\beta }\mathit{\iota } \end{gathered}$$

Hence, ${\mathit{e}}^{\mathbf{\gamma }\mathbf{\iota }}$ can be represented as ${\mathit{e}}^{\mathbf{\alpha }\mathbf{\iota }}\mathbf{\angle }\mathit{\beta }\mathit{\iota }$.

Therefore, the correct option is (a) True.