Obtain an expression for the electrostatic potential at a point in a space due to electric dipole consisting of charges +q at (0,0, a/2) and -q at (0,0,-a/2).

To derive the expression for the electrostatic potential at a point in space due to an electric dipole, we first need to understand the configuration of the dipole itself. An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance. In this case, the positive charge is located at (0, 0, a/2) and the negative charge at (0, 0, -a/2). The dipole moment, which is a vector quantity, can be defined as **p = q * d**, where **d** is the vector pointing from the negative charge to the positive charge. Here, **d** is along the z-axis, so we can express the dipole moment as **p = q * a * k**, where **k** is the unit vector in the z-direction.

Potential Due to Each Charge

The electrostatic potential **V** at a point in space due to a point charge is given by the formula:

V = \frac{k \cdot q}{r}

where **k** is Coulomb's constant, **q** is the charge, and **r** is the distance from the charge to the point where the potential is being calculated.

Calculating Distances

Let’s denote the point in space where we want to find the potential as **P(x, y, z)**. The distances from the point **P** to the charges can be calculated as follows:

• Distance to the positive charge (+q):

r_+q = √(x² + y² + (z - a/2)²)

• Distance to the negative charge (-q):

r_-q = √(x² + y² + (z + a/2)²)

Total Potential at Point P

The total potential at point **P** due to both charges is the algebraic sum of the potentials due to each charge:

V = V_{+q} + V_{-q}

Substituting the expressions for the potentials:

V = \frac{k \cdot q}{r_{+q}} - \frac{k \cdot q}{r_{-q}}

Now, substituting the distances we calculated earlier:

V = k \cdot q \left( \frac{1}{\sqrt{x^2 + y^2 + (z - a/2)^2}} - \frac{1}{\sqrt{x^2 + y^2 + (z + a/2)^2}} \right)

Expanding for Small Distances

In many practical situations, especially when the distance from the dipole is much larger than the separation of the charges (i.e., when **r >> a**), we can simplify this expression. We can use the binomial expansion for the square roots:

\sqrt{1 + x} \approx 1 + \frac{x}{2} \text{ for small } x

Applying this approximation, we can rewrite the potentials in a more manageable form, leading us to:

V \approx \frac{k \cdot p \cdot \hat{r} \cdot \hat{z}}{r^2}

where **p** is the dipole moment and **\hat{r}** is the unit vector pointing from the dipole to the point P.

Final Expression

Thus, the electrostatic potential at a point in space due to the dipole can be expressed as:

V \approx \frac{1}{4 \pi \epsilon_0} \frac{p \cdot \cos(\theta)}{r^2}

where **θ** is the angle between the dipole moment vector and the position vector from the dipole to the point P, and **r** is the distance from the dipole center to point P.

This expression captures the essence of the potential due to an electric dipole and highlights how it diminishes with distance, illustrating the unique nature of dipole fields compared to single point charges.