JEE Main & Advanced Physics Electrostatics & Capacitance Relation Between Electric Field and Potential

(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 it?s 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 \[\frac{volt}{meter}\].

(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 \[\vec{E}={{E}_{x}}\hat{i}+{{E}_{y}}\hat{j}+{{E}_{z}}\hat{k}\] 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 points in an electric field can be determined by knowing the boundary conditions \[dV=-\int_{{{r}_{1}}}^{{{r}_{2}}}{\overrightarrow{E\,}.\,\overrightarrow{dr}}=-\int_{{{r}_{1}}}^{{{r}_{2}}}{E.\,dr\,\cos \theta }\]