how do we know the electric field due to an electric dipole of dipole moment P at distance r from the centre of dipole is fiven by .. where theta is angle between p amd r vector..

To understand how we determine the electric field due to an electric dipole at a distance \( r \) from its center, we first need to grasp the concept of an electric dipole itself. An electric dipole consists of two equal and opposite charges, +q and -q, separated by a distance \( d \). The dipole moment \( \mathbf{P} \) is defined as the product of the charge and the distance between the charges, expressed as \( \mathbf{P} = q \cdot \mathbf{d} \). Now, let’s delve into how we derive the electric field at a point in space due to this dipole.

Understanding the Electric Field of a Dipole

The electric field \( \mathbf{E} \) created by a dipole can be derived using the principle of superposition, which states that the total electric field is the vector sum of the fields due to each charge. For a dipole, we consider the contributions from both the positive and negative charges at a point located at a distance \( r \) from the center of the dipole.

Mathematical Derivation

Let’s denote the position of the positive charge (+q) as \( \mathbf{r}_+ \) and the negative charge (-q) as \( \mathbf{r}_- \). The electric field due to a point charge is given by:

• For +q: \( \mathbf{E}_+ = \frac{1}{4\pi \epsilon_0} \frac{q}{|\mathbf{r} - \mathbf{r}_+|^2} \hat{\mathbf{r}}_+ \)
• For -q: \( \mathbf{E}_- = -\frac{1}{4\pi \epsilon_0} \frac{q}{|\mathbf{r} - \mathbf{r}_-|^2} \hat{\mathbf{r}}_- \)

Here, \( \hat{\mathbf{r}}_+ \) and \( \hat{\mathbf{r}}_- \) are the unit vectors pointing from the charges to the point where we are calculating the field. The total electric field \( \mathbf{E} \) at point \( r \) is then:

\( \mathbf{E} = \mathbf{E}_+ + \mathbf{E}_- \)

To simplify this expression, we can use the approximation valid for points far from the dipole (where \( r \) is much larger than \( d \)). In this case, we can express the distances as:

• \( |\mathbf{r} - \mathbf{r}_+| \approx r - \frac{d}{2} \cos \theta \)
• \( |\mathbf{r} - \mathbf{r}_-| \approx r + \frac{d}{2} \cos \theta \)

Substituting these approximations into the electric field equations and performing some algebra leads us to the expression for the electric field due to a dipole:

\( \mathbf{E} = \frac{1}{4\pi \epsilon_0} \frac{2\mathbf{P} \cos \theta}{r^3} \)

Interpreting the Result

In this equation, \( \theta \) is the angle between the dipole moment \( \mathbf{P} \) and the position vector \( \mathbf{r} \). The factor \( \cos \theta \) indicates that the electric field strength varies with the angle, being strongest along the axis of the dipole (where \( \theta = 0 \)) and weakest in the perpendicular direction (where \( \theta = 90^\circ \)).

This relationship highlights the directional nature of the electric field produced by a dipole, which is crucial in understanding phenomena such as molecular interactions and the behavior of dipoles in external electric fields.

Real-World Applications

Electric dipoles are not just theoretical constructs; they play a significant role in various fields, including chemistry, where molecular dipoles influence the polarity of molecules and their interactions. Understanding the electric field of dipoles helps in predicting how molecules will behave in electric fields, which is essential in fields like spectroscopy and material science.

In summary, the electric field due to an electric dipole at a distance \( r \) is derived from the contributions of both charges, simplified using approximations valid for large distances, and expressed in terms of the dipole moment and the angle \( \theta \). This understanding is foundational in both physics and chemistry, illustrating the interplay between charge distribution and electric fields.