Askiitians Tutor Team
Last Activity: 4 Months ago
To analyze the electric field generated by a tiny electric dipole at two far-off points, we need to consider the dipole moment and the nature of the electric field it produces. An electric dipole consists of two equal and opposite charges separated by a small distance, and its dipole moment \( \mathbf{p} \) is defined as the product of the charge and the distance between them, directed from the negative to the positive charge.
Understanding the Electric Field of a Dipole
The electric field \( \mathbf{E} \) due to a dipole at a point in space can be expressed in spherical coordinates, particularly when the point is far away from the dipole. The formula for the electric field at a distance \( r \) from the dipole is given by:
\[
\mathbf{E} \approx \frac{1}{4\pi \epsilon_0} \cdot \frac{2\mathbf{p} \cos \theta}{r^3}
\]
Here, \( \theta \) is the angle between the dipole moment vector and the line connecting the dipole to the point where the field is being calculated, and \( \epsilon_0 \) is the permittivity of free space.
Evaluating the Electric Fields at the Given Points
Now, let's evaluate the electric fields at the two specified points: \( (b, 0, 0) \) and \( (0, 0, b) \).
- Point (b, 0, 0): Here, the angle \( \theta \) between the dipole moment \( \mathbf{p} \) (which we can assume is along the z-axis for simplicity) and the x-axis is \( 90^\circ \). Thus, \( \cos(90^\circ) = 0 \). Therefore, the electric field at this point is:
\[
\mathbf{E}_{(b,0,0)} = 0
\]
- Point (0, 0, b): At this point, the angle \( \theta \) is \( 0^\circ \) since the dipole moment is aligned with the z-axis. Thus, \( \cos(0^\circ) = 1 \). The electric field at this point is:
\[
\mathbf{E}_{(0,0,b)} = \frac{1}{4\pi \epsilon_0} \cdot \frac{2\mathbf{p}}{b^3}
\]
Comparing the Electric Fields
From the calculations above, we see that:
- The electric field at point \( (b, 0, 0) \) is zero.
- The electric field at point \( (0, 0, b) \) has a non-zero magnitude and is directed along the z-axis.
Thus, the two electric fields are not equal in magnitude, nor are they equal in direction. In fact, one of them is zero while the other has a specific value. Therefore, the correct answer to the question is:
d) unequal in magnitude and opposite in direction
Conclusion
This analysis illustrates how the orientation of the dipole and the position of the observation points significantly affect the electric field's characteristics. Understanding these principles is crucial in fields like electrostatics and electromagnetism, where the behavior of electric fields is foundational.
