22.9
Electric Field of Two Equal and Opposite Charges
Atoms generally contain the same number of positively and negatively charged particles, protons, and electrons. Hence, they are electrically neutral.…
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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Q1: What is an electric dipole and how does it form?
An electric dipole consists of two equal and opposite charges separated by a distance. It forms when the centers of positive and negative charges do not coincide, such as in molecules like water or atoms where charge separation occurs. This configuration creates a weak, remnant electric field effect that depends on both the charge magnitude and separation distance.
Q2: Why do the electric field components cancel perpendicular to the charge separation?
At any point equidistant from both charges, the perpendicular field components have equal magnitude but opposite directions due to symmetry. When the two charges are swapped, the field in this perpendicular direction remains unchanged. This symmetry causes the perpendicular components to cancel, leaving only the field component along the separation direction.
Q3: How does the electric field of a dipole depend on distance?
At large distances from the dipole, the electric field is proportional to the product of charge and separation distance, and inversely proportional to the distance cubed. This inverse-cube dependence is weaker than the inverse-square dependence of individual charges. At very large distances, the dipole field approaches zero as the effects of the two charges cancel.
Q4: What role does symmetry play in calculating dipole electric fields?
Symmetry is a powerful tool for analyzing dipole fields. When a system looks identical under a particular operation, such as swapping the two charges, symmetry reveals which field components cancel and which reinforce. This approach simplifies calculations by identifying that perpendicular components cancel while parallel components add, directing the net field toward the negative charge.
Q5: Why is the dipole field weaker than individual charge fields despite stronger distance dependence?
Although the dipole field falls off as the inverse cube of distance, which is faster than the inverse square for individual charges, the overall field magnitude is weaker because it represents the combined effect of two opposite charges. The cancellation of perpendicular components and the partial cancellation of parallel components result in a net field that is comparatively weak, similar to why electrical and gravitational forces appear weaker at large distances.
Q6: How do you calculate the electric field at a point along the axis connecting the two charges?
Along the axis connecting the charges, both electric field components point in the same direction and reinforce each other. The individual fields from each charge add together, resulting in a net field twice as strong as from a single charge at that location. The direction is always toward the negative charge, and the magnitude depends on the charge value and separation distance.
Q7: Why do molecules like water exhibit electric fields despite being electrically neutral?
Although water molecules contain equal numbers of protons and electrons, making them electrically neutral overall, the centers of positive and negative charge do not coincide. This charge separation creates a dipole configuration with a weak, remnant electric field. The more the positive and negative charges separate within the molecule, the stronger the resulting electric field becomes.