Conditions for a E field to be possible

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To verify the existence of an electric field E in free space, it is essential to check that the divergence of E is zero (∇·E = 0). Additionally, for electrostatic systems, the curl of E must also be zero (∇×E = 0), confirming that the electric field is irrotational. Since the system in question is electrostatic, both conditions must be satisfied. Therefore, ensuring that the electric field is both irrotational and divergence-free is crucial for its validity in free space. This establishes the necessary conditions for the electric field's existence.
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If I want to verify an electric field E is possible in free space, do I just need to check ∇xE =0?
 
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In free space what always has to hold is ##\vec{\nabla}\cdot \vec{E} = 0## whereas ##\vec{\nabla}\times \vec{E} = 0## only holds in electrostatic systems. Is your system electrostatic?
 
Yes, the system is electrostatic.
 
Then yes you have to check that the electric field is irrotational. You also have to check that it is divergence-free.
 
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