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Electric Potential Due To An Electric Dipole

Electric Potential Due To An Electric Dipole :- The electric potential at a point in space due to an electric dipole is the algebraic sum of the potentials due to both charges.

Electric Potential Due To An Electric Dipole At Any Point in Space

(General Expression)

Let us consider a dipole with charges -q and +q placed at points A and B respectively. We have to calculate electric potential at a general point P distant r from the center O of the dipole. Let AP = r₁ and BP = r₂ . Draw normal AC and BD on PC.

Using superposition principle, total electric potential at point P :

$\displaystyle V=\frac{k(-q)}{{{r}_{1}}}+\frac{k(+q)}{{{r}_{2}}}=\frac{kq}{{{r}_{2}}}-\frac{kq}{{{r}_{1}}}$ …..(1)

Now in the above figure,

r₁ = AP ≅ PC = PO + OC = (r + a cosθ)

Similarly

r₂ = PB ≅ PD = PO – OD = (r – a cosθ)

Using these values in equation (1) we get,

$\displaystyle V=\frac{kq}{r-acos\theta }-\frac{kq}{r+acos\theta }$

$\displaystyle \Rightarrow V=kq\left[ \frac{\left( r+acos\theta \right)-\left( r-acos\theta \right)}{{{r}^{2}}-{{a}^{2}}co{{s}^{2}}\theta } \right]$

$\displaystyle \Rightarrow V=kq\left[ \frac{2acos\theta }{{{r}^{2}}-{{a}^{2}}co{{s}^{2}}\theta } \right]$

As dipole moment p = 2aq,

$\displaystyle \Rightarrow V=k\left[ \frac{pcos\theta }{{{r}^{2}}-{{a}^{2}}co{{s}^{2}}\theta } \right]$

For a short dipole r² >> ( a²cos²θ ) , hence

$\displaystyle V=\frac{kpcos\theta }{{{r}^{2}}}=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{pcos\theta }{{{r}^{2}}}$ …..(2)

As $\displaystyle pcos\theta =\overrightarrow{p}.\widehat{\text{ }r}$ , where $\displaystyle \widehat{\text{ }r}$ is a unit vector along the position vector $\displaystyle \overrightarrow{OP}=\overrightarrow{r}$ , so electric potential due to a short dipole (a << r) in vector form,

$\displaystyle V=\frac{1}{4\pi {{\varepsilon }_{0}}}\frac{\overrightarrow{p}.\widehat{\text{ }r}}{{{r}^{2}}}$ …..(3)