Derive an expression for electric potential at point due to an electric dipole.

Discuss the special cases.

To derive the expression for the electric potential at a point due to an electric dipole, we first need to understand what an electric dipole is. An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance, d. The dipole moment, denoted as **p**, is defined as the product of the charge and the separation distance: **p = q * d**. The electric potential (V) at a point in space due to this dipole can be derived using the principle of superposition.

Deriving the Electric Potential

Let's consider a point P located at a distance **r** from the midpoint of the dipole along the axis of the dipole. The potential at point P due to each charge can be expressed as:

• Potential due to +q: V_+q = \frac{kq}{r - \frac{d}{2}}
• Potential due to -q: V_-q = -\frac{kq}{r + \frac{d}{2}}

Here, **k** is Coulomb's constant, approximately equal to 8.99 x 10⁹ N m²/C². The total potential at point P is the sum of the potentials due to both charges:

V = V_+q + V_-q = \frac{kq}{r - \frac{d}{2}} - \frac{kq}{r + \frac{d}{2}}

To simplify this expression, we can find a common denominator:

V = kq \left( \frac{(r + \frac{d}{2}) - (r - \frac{d}{2})}{(r - \frac{d}{2})(r + \frac{d}{2})} \right)

This simplifies to:

V = \frac{kqd}{r^2 - \frac{d^2}{4}}

For points far away from the dipole, where **r** is much larger than **d** (i.e., r >> d), we can approximate the expression further:

V \approx \frac{kqd}{r^2}

Special Cases of Electric Potential

There are a couple of notable special cases to consider when discussing the electric potential due to a dipole:

• On the axial line: When the point is along the axis of the dipole, the potential simplifies to:

V_{axial} = \frac{2k \cdot p}{r^2}

• On the equatorial line: When the point is on the perpendicular bisector of the dipole (the equatorial line), the potential is given by:

V_{equatorial} = -\frac{k \cdot p}{r^2}

In these cases, the dipole moment **p** plays a crucial role in determining the potential at various points in space. The axial potential is positive, indicating that the potential is higher along the dipole's axis, while the equatorial potential is negative, showing that it is lower in that region.

Conclusion

In summary, the electric potential due to an electric dipole can be derived using the contributions from both charges, leading to a general expression that simplifies under certain conditions. Understanding these principles not only helps in grasping the behavior of electric dipoles but also lays the groundwork for more complex electrostatic concepts.