1.Derive an expression for the electroc field due to a diapole 2a at a point r from the diapole on the axial line.

2.draw a graph of electric field vs r greater than greater a.

3.if this diapole were kept in a uniform electric field E₀, diagrammatically represent the position of the diapole in stable and unstable equilibrium and write the expressions for the torque acting on the diapole in both the cases.

To derive the expression for the electric field due to a dipole at a point along the axial line, we first need to understand the configuration of the dipole. A dipole consists of two equal and opposite charges, +q and -q, separated by a distance 2a. The dipole moment, denoted as p, is defined as p = q × 2a and points from the negative charge to the positive charge.

Electric Field Due to a Dipole on the Axial Line

To find the electric field (E) at a point located at a distance r from the center of the dipole along the axial line, we can use the principle of superposition. The electric field due to a point charge is given by:

• E = k × |q| / r², where k is Coulomb's constant.

For the dipole, the electric field at point P due to the positive charge +q is:

• E₊ = k × q / (r - a)²

And the electric field due to the negative charge -q is:

• E₋ = -k × q / (r + a)²

The total electric field at point P is the vector sum of these two fields:

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

For points where r >> a (meaning r is much larger than a), we can use the binomial approximation to simplify the expression. This leads to:

• E ≈ (2kp) / r³

Thus, the electric field due to a dipole at a point on the axial line is:

Graph of Electric Field vs. Distance

When we plot the electric field (E) against the distance (r) for r greater than a, we observe that the electric field decreases with the cube of the distance. The graph will show a steep decline as r increases, indicating that the electric field strength diminishes rapidly as we move away from the dipole.

Dipole in a Uniform Electric Field

When a dipole is placed in a uniform electric field (E₀), it experiences torque, which tends to align the dipole with the field. The torque (τ) acting on the dipole is given by:

• τ = p × E₀

In terms of stability, there are two equilibrium positions for the dipole:

Stable Equilibrium

In stable equilibrium, the dipole aligns with the electric field. The torque acting on the dipole is:

• τ = -pE₀ cos(θ)

Here, θ is the angle between the dipole moment and the electric field. The negative sign indicates that the torque acts to restore the dipole to equilibrium when displaced.

Unstable Equilibrium

In unstable equilibrium, the dipole is oriented opposite to the electric field. The torque in this case is:

• τ = -pE₀ cos(θ)

However, since θ is 180 degrees (or π radians), the torque will act to increase the angle, pushing the dipole away from this position.

Visual Representation

To visualize this, you can draw two diagrams:

• For stable equilibrium, show the dipole aligned with the electric field lines.
• For unstable equilibrium, depict the dipole against the direction of the electric field lines.

In summary, the electric field due to a dipole on the axial line is inversely proportional to the cube of the distance from the dipole, and the dipole's behavior in a uniform electric field can be understood through the concepts of torque and equilibrium positions.