Relation between Electric Field and Electric Potential:
We have seen the relation between electric field and electric potential. With the help of this relation, we can evaluate potential difference between two points in known electric field.
In this section, our aim is to determine electric field for known potential function V.
For an electric field \(\vec E\), the small element \(\vec{dl}\) of displacement can be written in the form of derivative.
dV = \(-\vec{E} \cdot \overrightarrow{{d} l}\) = -Edl cosθ ………….. (1)
Where θ is the angle between electric field (\(\vec E\)) and length (\(\vec{dl}\))
\(-\frac{d V}{d l}\) = Ecosθ
The term \(-\frac{d V}{d l}\) represents decrease in potential with distance. It is clear from the above equation, that if the angle between 
and 
, θ = 0°, then the loss in potential with distance is maximum. In general form, the term is a scalar quantity but its maximum value \(\left(\frac{d V}{d l}\right)_{\max }\) in a particular direction (θ = 0) i. e.. in the direction of electric field. Thus, the maximum rate of loss in potential can be considered to be a vector which is along . In mathematical language, it is called potential gradient and it is represented as Grad V.
For an equipotential surface, the direction of Grad V is perpendicular to the surface. It can be understood with the help of figure 3.9. The equipotential surfaces are shown as S1and S2 on which the potential is V and V-dV respectively. The loss in potential in moving from point A on surface S1 to points B and C is dV, but the rate of change in potential with distance is \(\frac{d V}{A B}\) and \(\frac{d V}{A C}\) respectivelv,
which are different from each other.
Since, AB < AC
Thus, \(\frac{d V}{A B}>\frac{d V}{A C}\)
Now, surface AB is perpendicular to the surface. Thus, the rate of loss in potential is maximum in the direction normal to the surface.
Equation (1) can be written as:
where, E1 = Ecos1, component of \(\vec E\) along \(\vec {dl}\). Please note that we have used the symbols of partial derivatives which tells that the above equation in a definite axis (here for l axis) sums up the change in potential and component of E along that axis. If we assume l axis to be X, Y and Z axes, then the components of \(\vec E\) will be
This operator is called Del operator. With the help of equation (5), we can find out \(\vec E\) when the potential function V(x, y, z) is known.
If the potential function is a function of radius r. Then radial electric field
Er = \(-\frac{d V}{d r}\) ……………… (7)
