Electrostatic Forces Are Conservative
Electrostatic Forces Are Conservative :- Conservative force are those force in which work done by the force or against the force does not depend on the path followed but only depends on the initial and final positions. In other words the work done in closed path is zero.
To prove that electrostatic forces are conservative in nature, we will show that work done in moving a test charge +q0 over a closed path in the electrostatic filed is zero.
Work done in carrying the test charge +q0 from point A to B along path 1 :
Similarly, work done in carrying the test charge +q0 from point B to A along path 2 :
Total work done in carrying the test charge +q0 in closed path A → B → A :
![\displaystyle {{W}_{Total}}={{W}_{AB}}+{{W}_{BA}}=\left[ \left( {{V}_{B}}-{{V}_{A}} \right){{q}_{0}} \right]+\left[ \left( {{V}_{A}}-{{V}_{B}} \right){{q}_{0}} \right]](https://s0.wp.com/latex.php?latex=%5Cdisplaystyle+%7B%7BW%7D_%7BTotal%7D%7D%3D%7B%7BW%7D_%7BAB%7D%7D%2B%7B%7BW%7D_%7BBA%7D%7D%3D%5Cleft%5B+%5Cleft%28+%7B%7BV%7D_%7BB%7D%7D-%7B%7BV%7D_%7BA%7D%7D+%5Cright%29%7B%7Bq%7D_%7B0%7D%7D+%5Cright%5D%2B%5Cleft%5B+%5Cleft%28+%7B%7BV%7D_%7BA%7D%7D-%7B%7BV%7D_%7BB%7D%7D+%5Cright%29%7B%7Bq%7D_%7B0%7D%7D+%5Cright%5D&bg=ffffff&fg=000&s=0&c=20201002 "Electrostatic Forces Are Conservative 4")
Hence no work is done in moving the test charge +q0 over a closed path in an electrostatic filed. Hence electrostatic field is a conservative field and electrostatic forces are conservative forces in nature.
Mathematically,
i.e., line integral of electric field over a closed path in the electric filed is always zero.