To determine if the given expression represents an electrostatic field, we need to check if its curl is zero.

Question

To determine if the given expression represents an electrostatic field, we need to check if its curl is zero. An electrostatic field is a conservative field, and a fundamental property of conservative

ScanSolve AI Tutor · Accepted Answer

To determine if the given expression represents an electrostatic field, we need to check if its curl is zero. An electrostatic field is a conservative field, and a fundamental property of conservative fields is that their curl is zero ($
abla 	imes \vec{E} = 0$).

The given electric field is:
$$ \vec{E} = xy\hat{i} + 2yz\hat{j} + 3xz\hat{k} $$
Here, $E_x = xy$, $E_y = 2yz$, and $E_z = 3xz$.

Step 1: Calculate the curl of the electric field.
The curl operator is given by:
$$ 
abla 	imes \vec{E} = \left( \frac{\partial E_z}{\partial y} - \frac{\partial E_y}{\partial z} ight)\hat{i} + \left( \frac{\partial E_x}{\partial z} - \frac{\partial E_z}{\partial x} ight)\hat{j} + \left( \frac{\partial E_y}{\partial x} - \frac{\partial E_x}{\partial y} ight)\hat{k} $$

Step 2: Compute the partial derivatives for each component.
For the $\hat{i}$ component:
$$ \frac{\partial E_z}{\partial y} = \frac{\partial}{\partial y}(3xz) = 0 $$
$$ \frac{\partial E_y}{\partial z} = \frac{\partial}{\partial z}(2yz) = 2y $$
So, the $\hat{i}$ component is $0 - 2y = -2y$.

For the $\hat{j}$ component:
$$ \frac{\partial E_x}{\partial z} = \frac{\partial}{\partial z}(xy) = 0 $$
$$ \frac{\partial E_z}{\partial x} = \frac{\partial}{\partial x}(3xz) = 3z $$
So, the $\hat{j}$ component is $0 - 3z = -3z$.

For the $\hat{k}$ component:
$$ \frac{\partial E_y}{\partial x} = \frac{\partial}{\partial x}(2yz) = 0 $$
$$ \frac{\partial E_x}{\partial y} = \frac{\partial}{\partial y}(xy) = x $$
So, the $\hat{k}$ component is $0 - x = -x$.

Step 3: Combine the components to find the curl.
$$ 
abla 	imes \vec{E} = -2y\hat{i} - 3z\hat{j} - x\hat{k} $$

Step 4: Justify the answer.
Since $
abla 	imes \vec{E} = -2y\hat{i} - 3z\hat{j} - x\hat{k}$ is not equal to zero, the given expression for $\vec{E}$ does not represent an electrostatic field. An electrostatic field must have a curl of zero because it is a conservative field, meaning the line integral of the electric field around any closed loop is zero.

The expression below cannot be said to be that of an electrostatic field.
Justification: An electrostatic field is a conservative field, which implies that its curl must be zero. The calculated curl of the given field is $
abla 	imes \vec{E} = -2y\hat{i} - 3z\hat{j} - x\hat{k}$, which is not zero.

The final answer is $\boxed{	ext{No, the expression does not represent an electrostatic field.}}$

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