As the title states, Why is the Electric Field which is induced due to a time varying Magnetic Field Non Conservative in nature ?

Everywhere I read the answer that the line integral over a closed loop for an induced electric field is not zero hence it is non-conservative but my question is this. Why isn't the line integral over a closed loop zero for an induced electric field ?

PS : My knowledge about vectors is limited only to High school level, that is to say scalar,vector and scalar triple products. I haven't studied vector curl or divergence yet. Can it be explained upon these operations ?


1 Answer 1


I know you aren't familiar with vector calculus, but the answer is very simple in these terms. For an induced electric field we have $$\nabla\times\mathbf E=-\frac{\partial\mathbf B}{\partial t}$$

whereas for a conservative vector field it must be true that $$\nabla\times\mathbf E=0$$ which is what we have for electrostatic fields where $\mathbf B=0$ (and also enables us to define the scalar electric potential).

But for a little more intuition (but being less formal here), the induced electric field comes from changing magnetic fields, whose field lines forms loops. Therefore, it should be no surprise that these induced electric fields also form loops. Therefore, this points to why the work performed by the induced electric field in going around such a loop would be non-zero (since it is still the case that $\mathbf F=q\mathbf E$). You could somewhat think of it as being analogous to the work done by friction when you move an object around a closed loop, say on a rough table. Friction always points in the same direction relative to the path, so you can't get the work done by friction to be $0$.

  • $\begingroup$ Thanks. Can you also explain how it was determined that the curl of electrostatic electric fields is zero while that for induced electric fields is non zero ? Is it because induced electric fields form closed loops while electrostatic fields do not. Or has it been determined experimentally. $\endgroup$ Commented Nov 11, 2019 at 11:07
  • $\begingroup$ You can also explain it in terms of curl of you want. I read up a little on curl and divergence of vectors, so I have some idea as to what they are $\endgroup$ Commented Nov 11, 2019 at 11:18
  • $\begingroup$ @ShailendraSorout Coulomb's law is verified experimentally. You can show the electric field from Coulomb's law has no curl. $\endgroup$ Commented Nov 11, 2019 at 16:10
  • $\begingroup$ @ShailendraSorout Really all of Maxwell's equations have been experimentally validated. So we hold them to be true. $\endgroup$ Commented Nov 11, 2019 at 16:15

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