Let us assume that the dipole is at the centre of the one...

Let us assume that the dipole is at the centre of the one equipotential surface which is spherical (Fig.). On an equipotential surface the net electric field strength along the tangent of it becomes zero. Thus $-E_{0} \sin \theta+E_{\theta}=0 \quad$ or $-E_{0} \sin \theta+\frac{p \sin \theta}{4 \pi \varepsilon_{0} r^{3}}=0$ Hence $\quad r=\left(\frac{p}{4 \pi \varepsilon_{0} E_{0}}\right)^{1 / 3}$ Alternate : Potential at the point, near the dipole is given by, $\varphi=\frac{\vec{p} \cdot \vec{r}}{4 \pi \varepsilon_{0} r^{3}}-\vec{E}_{0} \cdot \vec{r}+$ constant, $=\left(\frac{p}{4 \pi \varepsilon_{0} r^{3}}-E_{0}\right) \cos \theta+$ Const For $\varphi$ to be constant, $\frac{p}{4 \pi \varepsilon_{0} r^{3}}-E_{0}=0$ or, $\frac{p}{4 \pi \varepsilon_{0} r^{3}}=E_{0}$ Thus $r=\left(\frac{p}{4 \pi \varepsilon_{0} E_{0}}\right)^{1 / 3}$

Let us assume that the dipole is at the centre of the one equipotential surface which is spherical (Fig.). On an equipotential surface the net electric field strength along the tangent of it becomes zero. Thus $-E_{0} \sin \theta+E_{\theta}=0 \quad$ or $-E_{0} \sin \theta+\frac{p \sin \theta}{4 \pi \varepsilon_{0} r^{3}}=0$
Hence $\quad r=\left(\frac{p}{4 \pi \varepsilon_{0} E_{0}}\right)^{1 / 3}$
Alternate : Potential at the point, near the dipole is given by, $\varphi=\frac{\vec{p} \cdot \vec{r}}{4 \pi \varepsilon_{0} r^{3}}-\vec{E}_{0} \cdot \vec{r}+$ constant,
$=\left(\frac{p}{4 \pi \varepsilon_{0} r^{3}}-E_{0}\right) \cos \theta+$ Const
For $\varphi$ to be constant,
$\frac{p}{4 \pi \varepsilon_{0} r^{3}}-E_{0}=0$ or, $\frac{p}{4 \pi \varepsilon_{0} r^{3}}=E_{0}$
Thus $r=\left(\frac{p}{4 \pi \varepsilon_{0} E_{0}}\right)^{1 / 3}$

Key Concepts

Electric Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance, characterized by a dipole moment which is a vector quantity representing both the magnitude of the charges and their separation. The dipole moment determines the strength and orientation of the dipole's electric field and potential in the surrounding space.

Dipole Potential

The potential due to an electric dipole at a point in space is given by an expression that involves the dipole moment, the distance from the dipole, and the cosine of the angle between the dipole moment and the position vector. This potential falls off with the cube of the distance from the dipole, highlighting the localized influence of the dipole compared to point charges.

Uniform Electric Field

A uniform electric field is one in which the electric field strength and direction remain constant throughout space. It introduces a linear potential variation over distance. When combined with the field of a dipole, the resulting electric potential and field patterns require the use of superposition, yielding complex configurations that are important in many applications such as studying induced dipole effects.

Equipotential Surface

Equipotential surfaces are regions in space where the electric potential has the same value, meaning no work is done when moving a charge along these surfaces. In an electric field configuration, the electric field is always perpendicular to the equipotential surfaces. Understanding these surfaces helps in visualizing and analyzing the behavior of electric fields, especially when different sources (like a dipole and a uniform field) are present.

Superposition Principle

The superposition principle states that in linear systems, the net electric potential or field due to multiple sources is the algebraic sum of the individual potentials or fields. This principle is particularly useful in cases where combined effects, such as those of a dipole and an external uniform field, are analyzed to determine resulting equipotential surfaces or field behaviors.

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