Problem 86 - Master Resource Book in JEE Main Physics

Question

Equipotentials at a great distance from a collection of charges whose total sum is not zero are approximately. \(\quad\) [NCERT Exemplar] (a) spheres (b) planes (c) paraboloids (d) ellipsoids

Step-by-Step Solution

Answer

Equipotentials are approximately spheres at a great distance from a non-zero sum of charges.

Step 1: Understanding the Problem

When we are looking at equipotentials at a great distance from a collection of charges, we are trying to determine the shape of these equipotential surfaces. These surfaces are where the electric potential is constant.

Step 2: Recognizing the Charges' Nature

If the total charge of the system is not zero, it implies that there is a net charge. The far field due to a net charge would behave similarly to a point charge.

Step 3: Analyzing Equipotential Surfaces Around Point Charges

For a point charge, equipotential surfaces are spheres. This is because, in 3D space, all points equidistant from the charge lie on a spherical surface, which is characteristic of how electric fields behave around point charges.

Step 4: Concluding Based on Theory

Therefore, even if the collection of charges is complex, at a large distance where the charges appear as a single net charge, the equipotential surfaces will be spherical. This is because their combined electric effect behaves like a monopole.

Key Concepts

Point Charge

A point charge is essentially a charge that is so small it can be considered as existing at a single point in space. We often use point charges when simplifying problems in electrostatics, especially because they make the mathematics a lot more manageable. The electric field created by a point charge is radial and emits outwards (if it’s a positive charge) or inwards (if negative). The strength of the electric field decreases as we move away from the charge.
An easy way to visualize the electric field around a point charge is to imagine it like the spokes of a wheel radiating out from a central hub. The further you go from the hub (the charge), the less intense these spokes become.
**Key Insights About Point Charges**

• The electric field \( E \) due to a point charge \( q \) at a distance \( r \) from it is given by \( E = \frac{kq}{r^2} \), where \( k \) is the Coulomb's constant.
• Point charges help us to approximate the behavior of charged objects at small scales.

Equipotential Surfaces

Equipotential surfaces are fascinating concepts in electrostatics. These are imaginary surfaces where every point has the same electric potential. For a single point charge, these surfaces are spherical in shape because all points at a certain distance from a charge have the same potential.
Imagine these surfaces as invisible layers surrounding the charge. No matter where you go on one of these layers, you won't gain or lose any electric potential energy moving around. Hence, if you're standing on an equipotential surface, it's like walking on a perfectly flat hilltop—no matter where you go, you're neither ascending nor descending.
**Facts About Equipotential Surfaces**:

• Equipotential surfaces are always perpendicular to electric field lines.
• For a point charge, these surfaces are spherical shells centered around the charge.
• On an equipotential surface, no work is done by the electric field on a charge; this implies any path between two points on the same equipotential surface requires zero work.

Electric Field Behavior

Electric fields provide crucial insights into the workings of electromagnetism. They represent the forces a charge would experience in space and are depicted by electric field lines. These lines start from positive charges and end on negative charges, representing the direction a positive test charge would move.
Understanding electric field behavior helps in visualizing how charges interact with each other. For example, the closer you are to a charge, the stronger the electric field and more tightly packed the lines are. This indicates stronger force interactions near the charge.
**Electric Field Properties**:

• Electric field lines never cross each other. Each point in space has a distinct electric potential and direction.
• The density of electric field lines indicates the field's strength. More lines close together mean a stronger field.
• An isolated charge (or net charge) behaves similarly to a monopole, where its electric field radiates uniformly in all directions, reminiscent of a star shedding light.