Question

Given the following electric field in Cartesian coordinates: \[ \mathbf{E} = x^2 y \hat{i} + y^2 z \hat{j} + z^2 x \hat{k}, \] which of the following statements is/are correct?

• A. The electric field is not conservative
• B. The electric field is static
• C. Both divergence and curl of the electric field are zero
• D. Electric field is neither conservative nor static

Hint:

To check if a vector field is conservative, calculate the curl. If the curl is zero, the field is conservative.

Answer 1

The Correct Option is A, B

Step 1: Check if the electric field is conservative. 
For a field to be conservative, its curl must be zero: \[ \nabla \times \mathbf{E} = \left( \frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z} \right) \times \left( x^2 y, y^2 z, z^2 x \right) \] By calculating the curl, we find that: \[ \nabla \times \mathbf{E} = (2xy - 2xz) \hat{i} + (2yz - 2xy) \hat{j} + (2zx - 2yz) \hat{k}. \] Since the curl is non-zero, the field is not conservative. 
Step 2: Check if the electric field is static. 
A static electric field should not depend on time. Since the given electric field has no time dependence, it is static. 
Step 3: Conclusion. Thus, the correct answer is (A) and (B).