Problem 79 - Arihant AIEEE Physics

Question

Identical charges of magnitude \(Q\) are placed at \((n-1)\) corners of a regular polygon of \(n\) sides each corner of the polygon is at a distance \(r\) from the centre. The field at the centre is: (a) \(\frac{k Q}{r^{2}}\) (b) \((n-1) \frac{k Q}{r^{2}}\) (c) \(\frac{n}{(n-1)} \cdot \frac{k Q}{r^{2}}\) (d) \(\frac{(n-1)}{n}: \frac{Q}{r^{2}}\)

Step-by-Step Solution

Answer

The field at the center is zero due to symmetry.

Step 1: Identifying the Problem

We begin by understanding that we have a regular polygon with \((n-1)\) equal charges, each of magnitude \(Q\), placed at its corners. Our task is to find the electric field at the center.

Step 2: Using Symmetry

A regular polygon of \(n\) sides shows symmetry around its center. If all corners except one have charges, the field due to multiple charges can be found considering their vectorial addition. Because polygons have symmetry, identical charges at symmetric points result in a zero net electric field due to pairs of charges cancelling each other out.

Step 3: Calculating Field from One Charge

The electric field \(E\) at the center due to one charge \(Q\) located a distance \(r\) away is given by Coulomb's law:\[ E_1 = \frac{kQ}{r^2} \]

Step 4: Net Field from Multiple Charges

For a polygon with \(n-1\) charges, pairing symmetry results in alternation and cancellation over the \(n\) polygon axes. Hence, the net field contribution of all such \((n-1)\) charges is zero due to perfect vector cancellations in their respective symmetric pairs.

Step 5: Conclusion

The overall effect of \((n-1)\) charges situated on a regular \(n\)-sided polygon results in zero net field at the center.

Key Concepts

Coulomb's law

Coulomb's law is fundamental in understanding electric fields. It describes the force between two point charges. The law states that the magnitude of the force between two charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them. Mathematically, it is expressed as:\[F = k \frac{|Q_1Q_2|}{r^2}\]where:

• \(F\) is the magnitude of the force between the charges,
• \(Q_1\) and \(Q_2\) are the magnitudes of the two charges,
• \(r\) is the distance separating the charges,
• \(k\) is Coulomb's constant \(\approx 8.99 \times 10^9 \text{ Nm}^2/\text{C}^2\).

For the exercise, you calculate the electric field due to a single charge using the formula:\[E = \frac{kQ}{r^2}\]This represents the field strength at a point in space due to a charge \(Q\). In our regular polygon problem, each charge contributes to the electric field at the center, initially computed using this principle. The key point for understanding the exercise is how this individual field contributes to or cancels out against others due to the symmetry of the setup.

symmetry

Symmetry plays a vital role in simplifying problems involving electric fields, especially in geometrical arrangements like regular polygons. In this specific exercise, the symmetry of a regular \(n\)-sided polygon determines how the electric fields due to multiple charges interact at the center.
The polygon's symmetry ensures that charges placed at equivalent positions relative to the center will have electric fields of equal magnitude but possibly different directions. For instance, in a square, charges at opposite corners generate fields that cancel out each other's influence at the center due to their vectors being equal and opposite.
This cancellation is why, in the problem, despite having \((n-1)\) charges, the net electric field at the center becomes zero due to perfect canceling from symmetric positions. Symmetry allows us to use vector addition more effectively and see that paired fields negate each other. By recognizing this, you can bypass extensive calculations and find easy solutions based on geometric layout.

vector addition

Vector addition is crucial in determining the resultant electric field from multiple charges. Vectors have both magnitude and direction, and understanding these properties helps you combine them accurately.
For electric fields, which are vectors, adding them involves considering both how strong the field is (magnitude) and where it points (direction). If two fields point in opposite directions, they will cancel each other out in terms of their net effect.

• Parallel Fields: Add their magnitudes if vectors are in the same direction.
• Opposite Fields: Subtract their magnitudes if vectors point in opposite directions.
• Perpendicular Fields: Use the Pythagorean theorem to find the resultant magnitude.

In our exercise, vector addition reveals that the net electric field at the center of a polygonal arrangement vanishes. This is because fields from symmetric charge pairs around the polygon are equal in magnitude but opposite in direction. Therefore, they cancel each other out, leaving us with zero net field. By learning how to apply vector addition, you can handle complex field setups more easily.