Is the Electric Field Always Conservative or Can it be Non-Conservative?

[Identity]

In some books I have seen:

$$\oint \mathbf{E} \cdot d\mathbf{s}=0$$

Since the Electric Field is meant to be conservative.


Elsewhere, however, I have also seen:

$$\oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt}$$


What's going on here?

Thanks

 

[Matterwave]

The first case is for electro-statics (no changing fields) and hence is true for the special case when $$\frac{d\Phi_B}{dt}=0$$

 

[Identity]

Ah, thanks Matterwave

 

[elect_eng]

The first case is for electro-statics (no changing fields) and hence is true for the special case when $$\frac{d\Phi_B}{dt}=0$$

This is true, of course.

What is amazing is that circuit or physics textbooks often quote the special case erroneously. They sometimes even call it a version of Kirchoff's Voltage Law, which is not correct. Any AC circuit that includes an inductor violates the quoted special case. Students should keep an eye out for this, and make sure the special case is really valid.

 

[sardar]

for your question. It is important to understand that both of these equations are correct, but they represent different scenarios. The first equation, \oint \mathbf{E} \cdot d\mathbf{s}=0, is known as the electric line integral and it applies in situations where the electric field is conservative. This means that the work done by the electric field on a charged particle moving along a closed path is independent of the path taken. In other words, the electric field does not have any "memory" of the path taken, only the initial and final positions of the particle.

On the other hand, the second equation, \oint \mathbf{E} \cdot d\mathbf{s} = -\frac{d\Phi_B}{dt}, is known as Faraday's law and it applies in situations where the electric field is non-conservative. This means that the work done by the electric field on a charged particle depends on the path taken. This can happen when there is a changing magnetic field present, which induces an electric field.

To summarize, the first equation applies in conservative situations where the electric field is "remembering" its source, while the second equation applies in non-conservative situations where the electric field is being induced by a changing magnetic field. Both equations are important in understanding and analyzing electric fields in different scenarios. I hope this helps clarify the confusion.