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Electric Field of Two Equal and Opposite Charges

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Consider a pair of source charges, one negative and the other positive, fixed along the x-axis and separated by a distance d. Let their midpoint be the origin. Consider a point P at a distance r from the origin.

The electric fields at P are directed toward minus-q and away from plus-q. Resolving the electric field along the x and y directions reveals their y-components have the same magnitude but opposite directions because P is equidistant from the two charges. Thus, the system possesses symmetry of the charges.

In the x-direction, the fields add up, resulting in twice the field due to each charge. Hence, the field is directed toward the negative charge.

At large distances compared to the separation, it is proportional to q-d and inversely proportional to the distance cubed, which is weaker than the individual fields.

Some molecules, like water, contain positive and negative charges separated by a distance. This expression helps calculate their fields.

At very large distances, it is zero, implying the effect of the two charges is canceled.

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$$\rightleftharpoonup{xx}$$ $$\longleftharp{xx}$$, $$\longrightharp{xx}$$,

Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral. However, the centers of the positive and negative charges do not always coincide. In such a scenario, the electric field of an atom may not be zero.

A separation of the positive and negative charges can lead to a weak, remnant effect of the positive and negative charges. The expectation is that the more the distance between the positive and negative charges, the more the electric field will be. The exact calculation can be facilitated by considering a positive and negative charge of equal magnitude separated by a distance. Such a system is called a dipole.

Symmetry is a powerful tool for calculating electric fields. When a system looks the same under a particular operation on it, it is said to be symmetric. As far as the electric field perpendicular to the separation direction is considered, an equidistant point from the two charges is symmetric under the operation: swapping the two charges. That is, if the two charges were swapped, the electric field in this direction would remain the same.

However, the field is not symmetrical in the direction of joining the two charges. The magnitudes and directions, both being the same, reinforce each other. Thus, at any point along the plane, there is a resulting electric field along the separation direction, directed towards the negative charge. As expected, its magnitude is proportional to the individual charges and the separation distance.

If the point is far removed from both the charges, the magnitude is approximately inversely proportional to the cube of its distance from the center of the charges. Thus, the dependence is stronger than the magnitude of the field due to individual charges. That is, it is a weaker field. That is why electrical forces appear weaker, although comparatively stronger than gravitational forces. They fall off sharply with the distance between objects.

Transcript

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