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Fellowroot
Why is the curl of an electric field E always equal to zero?
I think I figured this one out! Do you understand the electric field?
I'm not sure, but here's this: I think figured out not why the curl is always 0, but why the curl of the electric field from one source would be. The electric field force on another charge at any point has something to do with the electric field, here:|dw:1372802062313:dw||dw:1372802172444:dw|The vectors describe the force that the electron would have on a positive "test charge" that exists at the tail of the vector. So..... What is the curl? Rotation of the vectors. Infinitesimal turning, I guess. None of that! No curl! And look at the equation for curl: \[\nabla \times F=C\] The gradient is the direction of the greatest increase, and opposite in direction to greatest decrease, at a point. The force is in this same direction. The cross product, "\(\times\)," between two parallel vectors is zero!
There is one Stokes' theorem which says for a vector field ( in our case it is electric field ) \[\int\limits_{L}^{}E.dl = \int\limits_{S} (∇\times E) .ds\] The integrals are carried over closed loop L and the surface enclosing L which is S. Since electrostatic field is conservative. So its closed loop integral over any path will be zero. Therefore \[\int\limits_{L}^{}E.dl =0\] This gives \[\int\limits_{S} (∇\times E) .ds=0\] This should hold for every surface. Hence \[\ ∇\times E=0 \] This is NOT ALWAYS true. It is true only for electrostatic fields. For induced fields which are created by changing magnetic field \[∇\times E =-dB/dt\]
I don't follow that, but hopefully it helps! Haha!
curl is zero because electric field is a conservative field ^_^.. magnetic field on the other hand is non zero.. !! if you consider a point charge, then you ll see the dot product is zero everywhere.. but the situation holds good for all fields due TO STATIC CASES but induced fields on the other hands are non conservative.. and so the curl for induced electric fields are not zero.. that should be obvious magnetic field being non conservative cannot induce a conservative field..