the value of theta for which net electric field at a point p due to dipole moment vector p is

To determine the value of theta for which the net electric field at a point P due to a dipole moment vector **p** is zero, we first need to understand the configuration of an electric dipole and how the electric field behaves around it.

Understanding Electric Dipoles

An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance **d**. The dipole moment **p** is defined as:

• p = q * d

The direction of the dipole moment vector **p** points from the negative charge to the positive charge.

Electric Field Due to a Dipole

The electric field **E** at a point in space due to an electric dipole can be expressed in spherical coordinates, particularly in terms of the angle theta (θ) relative to the dipole axis. The formula for the electric field at a point along the axial line (the line extending from the positive charge through the negative charge) is given by:

• E = (1 / (4πε₀)) * (2p * cos(θ) / r³)

Here, **r** is the distance from the center of the dipole to the point P, and ε₀ is the permittivity of free space.

Finding the Condition for Zero Electric Field

To find the angle θ where the net electric field is zero, we need to consider the contributions from both the positive and negative charges. The electric field due to the positive charge points away from it, while the field due to the negative charge points towards it. At certain angles, these fields can cancel each other out.

The net electric field **E_net** at point P can be expressed as:

• E_net = E_positive - E_negative

For the electric field to be zero, we set:

• E_positive = E_negative

Calculating the Angle

From the dipole field equations, we know that:

• E_positive = (1 / (4πε₀)) * (p * (2cos(θ)) / r³)
• E_negative = (1 / (4πε₀)) * (p * (2cos(θ + π)) / r³)

Since cos(θ + π) = -cos(θ), we can rewrite the equation:

• E_negative = - (1 / (4πε₀)) * (p * (2cos(θ)) / r³)

Setting these equal gives us:

• (1 / (4πε₀)) * (p * (2cos(θ)) / r³) = - (1 / (4πε₀)) * (p * (2cos(θ)) / r³)

This leads us to conclude that the angle θ must be such that:

• cos(θ) = 0

The solutions to this equation occur at:

• θ = 90° and θ = 270°

Conclusion

Thus, the net electric field at point P due to a dipole moment vector **p** is zero when θ is either 90 degrees or 270 degrees. This means that at these angles, the contributions from the positive and negative charges of the dipole effectively cancel each other out, resulting in no net electric field at that point.