# Relation between Electric Field and Potential

## Relation between Electric Field and Potential

The electric filed exists only if there is an electric potential difference. If the charge is uniform at all point, however high the electric potential is, there will not be any electric field.

If the field is directed from lower potential to higher then the direction is taken to be positive. If the field is directed from higher potential to lower potential then then direction is taken as negative.

Relation between Electric Field and Potential:

$$E=-\frac{dV}{dr}$$; Where, E = Electric Field, V = Electric Potential and dr = Path Length

Where, negative sign is the electric gradient.

1) In an electric field rate of change of potential with distance is known as Potential Gradient.

2) Potential gradient is a vector quantity and its direction is opposite to that of electric field.

3) Potential Gradient relates with electric field according to the following relation, $$E=-\frac{dV}{dr}$$, this relation gives another unit of electric field is Volt/meter.

4) In the above relation negative sign indicates that in the direction of electric field potential decreases.

5) Negative of the slope of the V – r graph denotes intensity of electric field. i.e., $$\tan \theta =\frac{V}{r}=-E$$.

6) In space around a charge distribution we can also write $$\overrightarrow{E}={{E}_{x}}\widehat{i}+{{E}_{y}}\widehat{j}+{{E}_{z}}\widehat{k}$$.

Where, $${{E}_{x}}=-\frac{\partial V}{\partial x}$$ , $${{E}_{y}}=-\frac{\partial V}{\partial y}$$ and $${{E}_{z}}=-\frac{\partial V}{\partial z}$$.

7) With the help of formula: $$E=-\frac{dV}{dr}$$, Potential difference between any two point in an electric field can be determined by knowing the boundary condition.

$$dV=-\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{E.}}\,\,\overrightarrow{dr}=\int\limits_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{E.}}\,dr\,\cos \theta$$.