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Is it okay to say curl of electric displacement is zero in:

1) in electrostatics (curl of E is zero)

then followed by the following conditions:

2) when there is no polarization (curl of P is zero)

3) in uniform polarization (which makes curl of P is zero)

4) when there is no free charge presents (directly obtain electric displacement D is zero and thus curl of D is zero)

and one more, how does zero curl of electric displacement related with the symmetry?

because in Griffith 4th edition book says:

"

*If the problem exhibits spherical, cylindrical, or plane symmetry, then you can get*

**D**directly from equation by the usual Gauss's law methods. (Evidently in such cases**curl of P**is automatically zero)"