Suppose an electric field as depicted in the picture
$$E \propto y$$$$E = ky$$
Now when we take a charge $+q$ along the path $ABCDA$, Then :
$$W_{AB} = qky_2L$$$$W_{BC} = 0 \text{(Since, displacement is perpendicular to the field)}$$$$W_{CD} = -qky_1L$$$$W_{DA} = 0 \text{(Since, displacement is perpendicular to the field)}$$
Total work done :
$$W = W_{AB} + W_{BC} + W_{CD} + W_{DA}$$$$W = qky_2L + 0 - qky_1L + 0$$$$W = qkL (y_2 -y_1) $$
Now if $y_2 \neq y_1$, then $W \neq 0$ Which implies that such particular sort of electric field would act non-conservatively.
Since it can be easily asserted that such a field is impossible as it does not follow that static electric field must be conservative, I have done the derivative for a estabilised field of line charge here.
This field can be the E-component of an electromagnetic field that satisfies Maxwell's equations. Therefore, from a theoretical point of view this is a "possible electrical field". But, this field is not electro-static because it requires a time-variable B-field.
ReplyDeleteSee the answer at http://physics.stackexchange.com/questions/94512/non-conservative-behaviour-of-static-electric-field/95191?noredirect=1#comment194883_95191.
Thank you for your comment; But this post speculates that IF hypothetically a "static" field be established such as this, then it would be non-conservative and would pretty much violate energy conservation ! As of our current understanding of science and nature it is possible only with time varying magnetic fields, but again IF it could be made then previous assertions would become true.
Delete