Explanation:
Work done in an electric field by transporting an object having an electric charge q coulombs from a point having a potential V
1
volts to a point having potential V
2
is given by-
W=q(V
2
−V
1
)
The work done is also defined as the force×displacement.
W=
P
1
∫
P
2
Fds
=
P
1
∫
P
2
[q×E(x,y)]ds
=q×
P
1
∫
P
2
E(x,y)ds
=q×[V]
V
1
V
2
=q×[V
2
−V
1
]
as, E(x,y)=
ds
dV
Electric field at a point (x,y)=space gradient of electric potential.
So no matter whichever path S is taken from point P
1
to point P
2
, the integral will have only one value.