Consider the following conclusiond regarding the components of an electric field at a certain point in space given by `E_x = -Ky, E_y = Kx, E_z = 0` .

Options

A. The field is conservative.
B. The field is nonconservative.
C. The lines of force are straight lines.
D. The lines of force are circles.

Explanation

To solve the problem regarding the electric field components given by \( E_x = -Ky, E_y = Kx, E_z = 0 \), we need to determine whether this electric field is conservative. A field is conservative if the curl of the field is zero. Let's go through the steps to find the curl of the electric field.

### Step 1: Write down the electric field components
The electric field components are given as:
- \( E_x = -Ky \)
- \( E_y = Kx \)
- \( E_z = 0 \)

### Step 2: Set up the curl of the electric field
The curl of a vector field \( \mathbf{E} \) is given by the formula:
\[ \nabla \times \mathbf{E} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ E_x & E_y & E_z \end{vmatrix} \]
Substituting the components of the electric field:
\[ \nabla \times \mathbf{E} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ -Ky & Kx & 0 \end{vmatrix} \]

### Step 3: Calculate the determinant
Calculating the determinant, we expand it as follows:
\[ \nabla \times \mathbf{E} = \hat{i} \left( \frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(Kx) \right) - \hat{j} \left( \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(-Ky) \right) + \hat{k} \left( \frac{\partial}{\partial x}(Kx) - \frac{\partial}{\partial y}(-Ky) \right) \]

### Step 4: Evaluate each term
1. For the \( \hat{i} \) component:
\[ \frac{\partial}{\partial y}(0) - \frac{\partial}{\partial z}(Kx) = 0 - 0 = 0 \]

2. For the \( \hat{j} \) component:
\[ \frac{\partial}{\partial x}(0) - \frac{\partial}{\partial z}(-Ky) = 0 - 0 = 0 \]

3. For the \( \hat{k} \) component:
\[ \frac{\partial}{\partial x}(Kx) - \frac{\partial}{\partial y}(-Ky) = K - (-K) = K + K = 2K \]

### Step 5: Combine the results
Thus, we find:
\[ \nabla \times \mathbf{E} = 0 \hat{i} - 0 \hat{j} + 2K \hat{k} = 2K \hat{k} \]

### Step 6: Conclusion
Since the curl of the electric field \( \nabla \times \mathbf{E} \) is not zero (it is \( 2K \hat{k} \)), the electric field is not conservative.

### Final Answer
The electric field is not conservative.

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