# for a conservative field ….....,,,,,.,.,. .,......                ∫E.dl=? .. and why

Arun
25750 Points
6 years ago

The vanishing of closed line integrals means that the field is conservative.

Since∮→E⋅d→l" id="MathJax-Element-8-Frame" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0">

→ld⋅→E∮
is equivalent to→∇×→E=0" id="MathJax-Element-9-Frame" role="presentation" style="box-sizing: border-box; margin: 0px; padding: 0px; border: 0px; font-variant: inherit; font-stretch: inherit; line-height: normal; font-family: inherit; vertical-align: baseline; display: inline; word-spacing: normal; word-wrap: normal; white-space: nowrap; float: none; direction: ltr; max-width: none; max-height: none; min-width: 0px; min-height: 0px; position: relative;" tabindex="0">

0=→E×→∇
, the "physical interpretation" is the the electric field is irrotational, i.e. it has no "vortices". The, more valuable, mathematical implication is that there is a scalar potential whose gradient is the electric field.

Arun
25750 Points
6 years ago
Electrostatic forces are conservative in nature.When a unit positive test charge is moved from one point to another against a electrostatic field of positive line charge it undergoes repulsion due to like charges.Hence a certain amount of work is done to move the charge towards the positive line charge.When an external force does work in taking a body from one point to another against a force , then work gets stored as potential energy.Once when the force is removed the body moves , gaining kinetic energy and losing equal amount of potential energy .Thus the sum of kinetic and potential energy is conserved.Thus no work is done in moving a unit positive test charge over a closed path in an electric field.Mathematically ,we express this result as ,line integral of closed path in the electric field is always