Question

Prove that E=– grad V

Answer 1

Answer:

As we know, the potential at any point due to a point charge q is given by V=kq/r , where k is known as the constant of proportionality whose value is 9 x 10^9 Nm^2/C^2 and r is the distance between the point and the charge q.

Now taking del operator on both sides of the equation V=kq/r

$grad \: v = kq \: grad (\frac{1}{r} )$

$⟹grad \: v = kq( - \frac{r}{ {r}^{2} } )$

$⟹ - grad \: v = (k \frac{q}{ {r}^{2} }r )....(1)$

also we know,

Electric field

$E = k \frac{q}{ {r}^{2} } r...(2)$

hence,

E= - grad v