Question

show that the electric field at any point is equal to negative potential gradient at the point​

Answer 1

Answer:

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Answer 2

Given:

Electric field at any point

To Show : Electric field at any point is equal to negative potential gradient at the point​.

Solution :

Standard formula of electric field intensity and potential are given by

$\[E{\text{ }} = {\text{ }}\frac{{KQ}}{{{r^2}}}\]$ , $\[V{\text{ }} = {\text{ }}\frac{{KQ}}{r}\]$

here,

Q is the charge, r is the distance between charge and observation point

$\[V{\text{ }} = {\text{ }}\frac{{KQ}}{r}\]$

differentiate with respect to r

$\[\frac{{dV{\text{ }}}}{{dr}} = {\text{ }} - \frac{{KQ}}{{{r^2}}} = {\text{ }} - E\]$

$\[dV{\text{ }} = {\text{ }} - E.dr\]$

$\[E{\text{ }} = {\text{ }} - \frac{{dv}}{{dr}}\]$

The quantity  $\[ - \frac{{dv}}{{dr}}\]$  , is the rate of change of potential with distance and is called potential gradient.

Hence, Electric field at any point is equal to negative potential gradient at the point​.