Name: Electric Dipole in a Non-uniform Electric Field
Uploaded: 2019-07-19
Duration: 3 min 23 s
Description: A dipole experiences no net force in a uniform electric field. Will it exhibit similar behaviour in a non-uniform electric field? Stay tuned to find out!

Question

Define electric potential and obtain an expression between intensity of electric field due to a point charge and electric potential. Also show that electric field is gradient of electric potential.

Accepted Answer

Electric potential $\to$ It is defined as work done per unit test charge.

$i . e ., [V = \frac{W}{q}]$

Expression for electric field intensity at a point due to point charge $\to$

By Coulomb's' law

$F = \frac{1}{4 π \in_{0}} \frac{q_{1} q_{2}}{r^{2}} F = \frac{1}{4 π \in_{0}} \frac{{qq}_{0}}{r^{2}} (q_{0} = 1 c) ∴ F \frac{1}{4 π \in_{0}} \frac{q}{r^{2}}$

This force acting on unit + ve charge is called intensity

So $[E = \frac{1}{4 π \in_{0}} \frac{q}{r^{2}}]$

Expression for electric potential at a point due to point charge $\to$

By coulombs' law

$F = \frac{1}{4 π \in_{0}} \frac{q_{1} q_{2}}{x^{2}} F = \frac{1}{4 π \in_{0}} \frac{q q_{0}}{x^{2}} (q_{0} = 1 c)$

Total work done is

$W = \int_{\propto}^{r} - Fdx = \int_{\propto}^{r} - \frac{1}{4 π \in_{0}} \frac{q}{x^{2}} dx = \frac{- q}{4 π \in_{0}} \int_{\infty}^{r} x^{- 2} dx = \frac{- q}{4 π \in_{0}} {[\frac{x^{- 1}}{- 1}]}^{r}_{\infty} W = + \frac{q}{4 π \in_{0}} {[\frac{1}{x}]}^{r}_{\infty} = \frac{q}{4 π \in_{0}} [\frac{1}{r} - \frac{1}{\infty}] ∴ W = \frac{1}{4 π \in_{0}} \frac{q}{r} (\frac{1}{\propto} = 0) ∴ W = v so [v = \frac{1}{4 π \in_{0}} \frac{q}{r}]$

To show the electric field is gradient of electric potential $\to$

The small work done in displacing $q_{0}$ from $P$ to $P^{1}$ is

$dw = - Fdr = - q_{0} Edr or \frac{dw}{q_{0}} = - Edr or dv = - Edr [E = - \frac{dv}{dr}]$

i.e., the electric intensity is the gradient of electric potential.

Define electric potential and obtain an expression between intensity of electric field due to a point charge and electric potential. Also show that electric field is gradient of electric potential.

Important Questions on Electric Potential

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