Derive an expression for the electric field intensity at a point on the equatorial line of an electric dipole of dipole moment P and length 2a. What is the direction of this field?

To derive the expression for the electric field intensity at a point on the equatorial line of an electric dipole, we first need to understand the configuration of the dipole and the nature of the electric field it produces. An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance of 2a. The dipole moment, denoted as P, is defined as the product of the charge and the distance between the charges: P = q * 2a.

Understanding the Electric Field of a Dipole

For an electric dipole, the electric field at a point in space can be derived using the principle of superposition. The electric field due to a dipole at a point in space is the vector sum of the electric fields due to each charge.

Configuration of the Dipole

Consider a dipole oriented along the z-axis, with the positive charge located at (0, 0, a) and the negative charge at (0, 0, -a). The point where we want to calculate the electric field is located on the equatorial line, which lies in the xy-plane, at a distance r from the center of the dipole (the midpoint between the charges).

Calculating the Electric Field

The electric field due to a point charge is given by the formula:

• E = k * |q| / r²

where k is Coulomb's constant. For our dipole, we need to calculate the electric fields due to both charges at the point on the equatorial line.

Let’s denote the distance from the center of the dipole to the point on the equatorial line as r. The distance from the positive charge to this point is:

• r + a

And the distance from the negative charge is:

• r - a

The electric field due to the positive charge (+q) at the point is:

• E₊ = k * q / (r + a)²

The electric field due to the negative charge (-q) at the same point is:

• E₋ = k * q / (r - a)²

Net Electric Field Calculation

Since the electric fields due to the positive and negative charges are in opposite directions along the equatorial line, the net electric field (E) at that point can be expressed as:

• E = E₊ - E₋

Substituting the expressions for E₊ and E₋, we have:

• E = k * q / (r + a)² - k * q / (r - a)²

To simplify this expression, we can factor out k * q:

• E = k * q * [1 / (r + a)² - 1 / (r - a)²]

Now, applying the difference of squares formula:

• 1 / (r + a)² - 1 / (r - a)² = [(r - a)² - (r + a)²] / [(r + a)² * (r - a)²]

After simplification, we find that:

• E = -\frac{2kq(2a)}{r^4} = -\frac{2kP}{r^3}

Direction of the Electric Field

The direction of the electric field on the equatorial line is perpendicular to the dipole axis and points away from the dipole. This is because the positive charge creates a field that points away from it, while the negative charge creates a field that points towards it. The net effect results in a field that is directed outward from the dipole's center along the equatorial line.

Final Expression

In conclusion, the electric field intensity at a point on the equatorial line of an electric dipole with dipole moment P and length 2a is given by:

• E = -\frac{2kP}{r^3}

This expression highlights how the electric field strength decreases with the cube of the distance from the dipole, illustrating the rapid fall-off of the electric field strength as one moves away from the dipole.