Find the curl and divergence of each of the following vector fields:- **\[ \vec{E} = \hat{x} A e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{x}. \] Assume that $ A, k_0, \omega, $ and $ t $ are all independent of $ x, y, $ and $ z. $**- **\[ \vec{D} = \hat{x} A e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{y}. \] Assume that $ A, k_0, \omega, $ and $ t $ are all independent of $ x, y, $ and $ z. $**- **\[ \vec{H} = (\hat{z} A + \hat{y} B) e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{z}. \] Assume that $ A, B, k_0, \omega, $ and $ t $ are all independent of $ x, u, $ and $ z. $**

The position vector is : r→=xx^+yy^+zz^ The curl of the given vectors is :…

Answered: Find the curl and divergence of each of…

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Question

Find the curl and divergence of each of the following vector fields:

- **\[ \vec{E} = \hat{x} A e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{x}. \] Assume that $ A, k_0, \omega, $ and $ t $ are all independent of $ x, y, $ and $ z. $**

- **\[ \vec{D} = \hat{x} A e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{y}. \] Assume that $ A, k_0, \omega, $ and $ t $ are all independent of $ x, y, $ and $ z. $**

- **\[ \vec{H} = (\hat{z} A + \hat{y} B) e^{i \vec{k} \cdot \vec{r} - \omega t}, \] where \[ \vec{k} = k_0 \hat{z}. \] Assume that $ A, B, k_0, \omega, $ and $ t $ are all independent of $ x, u, $ and $ z. $**

Expert Solution

Step 1

The position vector is : $\vec{r} = x \hat{x} + y \hat{y} + z \hat{z}$

The curl of the given vectors is :

$\vec{\nabla} \times \vec{E} = |\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A e^{i (x k_{0} - ω t)} & 0 & 0 \end{array}| = 0$

$\vec{\nabla} \times \vec{D} = |\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ A e^{i (y k_{0} - ω t)} & 0 & 0 \end{array}| = - A i k_{0} e^{i (y k_{0} - ω t)} \hat{z}$

$\vec{\nabla} \times \vec{H} = |\begin{array}{ccc} \hat{x} & \hat{y} & \hat{z} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ 0 & B e^{i (z k_{0} - ω t)} & A e^{i (z k_{0} - ω t)} \end{array}| = - B i k_{0} e^{i (z k_{0} - ω t)} \hat{x}$

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