find the locus of all points where the resultant E field will always have a bearing of 45degree with the axis of short dipole.

To find the locus of all points where the resultant electric field (E field) makes a 45-degree angle with the axis of a short dipole, we need to delve into the characteristics of the electric field generated by a dipole and the geometric implications of this angle.

Understanding the Electric Field of a Dipole

A short dipole consists of two equal and opposite charges separated by a small distance. The electric field produced by a dipole at a point in space can be described using the dipole moment, which is defined as:

• p = q * d,

where q is the magnitude of each charge and d is the separation distance between them. The electric field due to a dipole at a point in space can be expressed in spherical coordinates (r, θ) as:

• E(r, θ) = (1/4πε₀) * (2p * cos(θ) / r³) in the axial direction,
• E(r, θ) = (1/4πε₀) * (p * sin(θ) / r³) in the equatorial direction.

Finding the Angle Condition

For the resultant electric field to make a 45-degree angle with the axis of the dipole, we can use the tangent of the angle:

• tan(45°) = 1.

This means that the magnitudes of the axial and equatorial components of the electric field must be equal:

• E_axial = E_equatorial.

Setting Up the Equation

From the expressions for the electric field components, we can set up the equation:

• (2p * cos(θ) / r³) = (p * sin(θ) / r³).

By simplifying this, we can cancel out common terms:

• 2 * cos(θ) = sin(θ).

Now, rearranging gives us:

• sin(θ) = 2 * cos(θ).

Using Trigonometric Identities

We can use the identity tan(θ) = sin(θ)/cos(θ) to express this in terms of tangent:

• tan(θ) = 2.

This implies that:

• θ = arctan(2).

Identifying the Locus of Points

The angle θ represents the angle from the dipole axis. The locus of points where the resultant electric field makes a 45-degree angle with the dipole axis can be visualized as a cone extending from the dipole. The angle of the cone is determined by the angle we found, which corresponds to the direction where the electric field is equal in both components.

Geometric Interpretation

In three-dimensional space, this cone will have its vertex at the center of the dipole and will extend outward. The surface of this cone represents all the points where the resultant electric field maintains that 45-degree angle with respect to the dipole axis. The radius of the cone will increase as you move away from the dipole, but the angle will remain constant.

Conclusion

In summary, the locus of all points where the resultant electric field of a short dipole makes a 45-degree angle with the dipole axis forms a conical surface. This cone has its apex at the dipole's center and extends outward, maintaining the specified angle throughout its surface. Understanding this concept helps in visualizing electric fields and their interactions in electrostatics.