Problem 50 - Arihant AIEEE Physics

Question

A point charge \(q\) and a charge \((-q)\) are placed at \(x=-a\) and \(x=+a\) respectively. Which of the following represents a part of \(E-x\) graph ?

Step-by-Step Solution

Answer

The graph of \( E \) vs \( x \) is anti-symmetric about the origin, with zero at \( x = 0 \).

Step 1: Understanding the Problem

We have two point charges: charge \( q \) at position \( x = -a \) and charge \( -q \) at position \( x = +a \). We need to represent the electric field \( E \) as a function of position \( x \) on a graph.

Step 2: Considering Symmetry and Direction

The electric field due to a positive charge \( q \) points away from the charge, while the field due to a negative charge \( -q \) points towards the charge. Because of symmetry and opposite charges, the field will be symmetrically opposite across the y-axis with zero potential at \( x=0 \).

Step 3: Finding Electric Field at Different Regions

1. For \( x < -a \), the field \( E \) is predominantly influenced by charge \( q \) at \( x = -a \) and points left.2. For \( -a < x < a \), between the charges, fields from \( q \) and \( -q \) add because they point in opposite directions towards each other.3. For \( x > a \), the field is predominantly influenced by charge \( -q \) at \( x = +a \) and points right.

Step 4: Determining Electric Field Strengths

The electric field due to a point charge is given by \( E = \frac{kq}{r^2} \). So:- For \( x < -a \), the field is negative and strong.- At \( x = 0 \), \( E = 0 \) due to the cancellation of the fields.- For \( x > a \), the field is positive but weaker with increasing \( x \).

Step 5: Analyzing the Graph Shape

The graph of the electric field \( E \) vs \( x \) will show:- Negative values for \( x < -a \), decreasing with decreasing \( x \).- A zero crossing at \( x = 0 \). - Positive values for \( x > a \), decreasing as \( x \) increases.The graph should be anti-symmetric around the origin.

Key Concepts

Point Charge

A point charge is a fundamental concept in electromagnetism describing a charge located at a single point in space. This is an idealization that helps simplify calculations of electric fields.

• Point charges can be either positive or negative. A positive charge causes electric fields that point away from the charge, while a negative charge attracts electric field lines towards itself.
• The magnitude of the electric field \'E\' produced by a point charge diminishes with the square of the distance from the charge, based on Coulomb's Law: \( E = \frac{kq}{r^2} \), where \( k \) is Coulomb's constant, \( q \) is the magnitude of the charge, and \( r \) is the distance from the charge to the point of interest.

Consider the example in our exercise: two opposing charges are placed symmetrically about the origin. This creates an intriguing situation to explore how electric fields from point charges interact.

Symmetry in Electric Fields

Symmetry plays a crucial role in understanding and visualizing electric fields created by multiple charges. In the given scenario, we have two equal yet opposite charges positioned symmetrically at \( x = -a \) and \( x = +a \).

• Due to this symmetry and the nature of electric fields from positive and negative charges, the resulting electric field configuration is symmetric about the y-axis.
• At the midpoint between the two charges (\( x=0 \)), the electric fields from each charge exactly cancel out, yielding a net electric field of zero.

Recognizing this symmetric property helps in predicting the behavior of the electric field graph, as well as simplifying calculations by reducing the complexity of each section of the graph.

Electric Field Strength

Electric field strength is a measure of the force experienced per unit charge in an electric field. The formula \( E = \frac{kq}{r^2} \) operates as a cornerstone to calculate the strength at specific positions.

• For \( x < -a \), the electric field is dominated by the charge at \( x = -a \). Its influence weakens as we move farther left away from the charge.
• Within the region \( -a < x < a \), electric fields from both charges overlap. These fields add up since they point in opposite directions towards each other, forming a strong field.
• Past \( x = +a \), the influence of the negative charge at \( x = +a \) prevails. Here, the field strength decreases with increasing \( x \) distance from the charge.

These varying strengths across different regions result in an anti-symmetric graph of the electric field vs. x, showcasing strong interactions between the charges and their fields.