Problem 93 - A Complete Resource Book in Physics for JEE Main

Question

Four charges \(+2 q,-2 q,-3 q\), and \(+3 q\) are kept in the corners of a square of side \(a\). The total field at the centre \(O\) is, (A) Zero (B) \(\frac{2 \sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\) (C) \(\frac{\sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\) (D) \(\frac{10 \sqrt{2} q}{4 \pi \varepsilon_{0} a^{2}}\)

Step-by-Step Solution

Answer

The net electric field at the center O of the square (considering directions) is given by \(E_{net} = \frac{\sqrt{2}q}{4 \pi \varepsilon_{0} a^{2}}\), which corresponds to option (C).

Step 1: Name the Charges and Points

Let the charges be A (+2q), B (-2q), C (-3q), and D (+3q). Name the center of the square O.

Step 2: Calculate the Distance from Each Charge to Point O

From the problem, we know that the side of the square is "a". Use the Pythagorean theorem to find the distance from each corner to point O (the center). Distance AO = BO = CO = DO = \(\sqrt{(\frac{a}{2})^{2}+(\frac{a}{2})^{2}} = \frac{a \sqrt{2}}{2}\).

Step 3: Calculate the Electric Field at Point O Due to Each Charge

Use the electric field formula: \(E = \frac{kQ}{r^2}\), where k = \(\frac{1}{4 \pi \varepsilon_{0}}\). Calculate the electric field at point O due to each charge keeping the respective directions in mind.

Step 4: Determine the Net Electric Field at Point O by Adding the Electric Fields Produced by the Charges

Add the electric fields produced by the charges: \(E_{net} = E_A + E_B + E_C + E_D\).

Step 5: Compare the Calculated Net Electric Field to the Given Choices and Choose the Correct Answer

Comparing the calculated net electric field at point O to the given choices, choose the correct answer.

Key Concepts

Coulomb's Law

Coulomb's Law is essential in understanding electric fields. It describes the force between two point charges. This force is directly proportional to the product of their charges and inversely proportional to the square of the distance between them.

The formula is given by:

• \( F = \frac{k |q_1 q_2|}{r^2} \)

Where:

• \( F \) is the force between the charges
• \( q_1 \) and \( q_2 \) are the magnitudes of the charges
• \( r \) is the distance between the charges
• \( k \) is Coulomb's constant, \( \frac{1}{4 \pi \varepsilon_0} \)

This law helps us calculate the electric field generated by point charges in various configurations. The electric field due to a single charge is:

• \( E = \frac{kQ}{r^2} \)

Here, \( E \) is the electric field at a distance \( r \) from the charge \( Q \). Understanding how electric fields behave under Coulomb's Law is crucial when dealing with multiple charges.

Charge Distribution

When dealing with problems involving multiple charges, understanding charge distribution is key. Here, charges are placed at specific positions, affecting the resulting electric field. In our problem, we have four charges at the corners of a square. The arrangement of these charges and their magnitudes determine the field outcome at the center.

Important points about charge distribution include:

• Symmetry of the configuration. This can sometimes simplify calculations, as symmetrical distributions might cancel out some effects.
• The magnitude and sign of each charge as they influence the direction and strength of the electric field produced.
• The exact placement of charges relative to each other and the point of interest, which in our case is the center of the square.

In analyzing such configurations, one often uses symmetry and vector addition of electric fields to determine the net field at a point.

Superposition Principle

The superposition principle is a fundamental concept when dealing with multiple electric fields. It states that the total electric field created by multiple sources is the vector sum of the individual fields from each source. This principle simplifies complex problems by allowing us to calculate the effect of each charge individually and then add them up. In our exercise:

• Calculate the electric field due to each charge using Coulomb's law.
• Consider the direction each field is acting. Some fields might have components that cancel each other out or add up depending on their directions.
• Add the vector fields to find the total electric field at the center of the square.

By using superposition, the complex interplay of forces becomes more manageable, making it possible to predict the net electric field more accurately.