Problem 24 - Arihant AIEEE Physics

Question

A negatively charged metallic ball is supported on a rigid insulating stand. We wish to measure the electric field \(E\) at a point \(P\) in the same horizontal level as that of the metallic ball. To do so, we put a positive charge \(q_{0}\) and measure \(F / q_{0}\). The electric field at the point \(P\) is: (a) \(=\frac{F}{q_{0}}\) (b) \(<\frac{F}{a_{0}}\) (c) \(>\frac{F}{q_{0}}\) (d) none of these

Step-by-Step Solution

Answer

The electric field \( E \) at point \( P \) is \( E = \frac{F}{q_0} \), so the answer is (a).

Step 1: Understanding the Problem

We need to determine the correct expression for the electric field, \( E \), at point \( P \) related to the force \( F \) experienced by a test charge \( q_0 \). We are given several options that describe relationships between \( E \) and \( \frac{F}{q_0} \).

Step 2: Static Electric Field Definition

The electric field \( E \) at a point \( P \) due to a charge is defined as \( E = \frac{F}{q_0} \), where \( F \) is the force experienced by the test charge \( q_0 \) placed at that point. This is the direct definition from Coulomb's Law for electric fields.

Step 3: Analyzing the Options

Given the definition \( E = \frac{F}{q_0} \), we can eliminate options (b) and (c) because they suggest inequalities. Option (a) states \( E = \frac{F}{q_0} \), which directly matches the definition we established. Option (d) can also be eliminated as it suggests none of the above options are correct.

Step 4: Selecting the Correct Option

From the analysis in the previous steps, the option that correctly describes \( E \) is based on the direct definition of electric field from force and charge. Thus, option (a) is the correct answer.

Key Concepts

Coulomb's Law

Coulomb's Law is essential when calculating the electric force between two charged objects. It states that the force (\( F \)) between two point charges is directly proportional to the multiplicative product of the magnitudes of charges (\( q_1 \) and \( q_2 \)), and inversely proportional to the square of the distance (\( r \)) between them. The law can be expressed mathematically as:

• \( F = k \frac{|q_1 q_2|}{r^2} \)

where \( k \) is Coulomb's constant, approximately \( 8.99 \times 10^9 \, ext{N m}^2/ ext{C}^2 \).
This law helps us understand that as charges move farther apart, the force decreases with the square of the distance. Besides, the nature of the force can either be attractive or repulsive depending on the types of charges involved:

• Attractive Force: If the charges are of opposite signs.
• Repulsive Force: If the charges are of the same sign.

Understanding Coulomb's Law is foundational when dealing with electric fields and charges because it explains how they interact with one another.

Test Charge

A test charge is a small positive charge used to measure the intensity and direction of an electric field at a specific point. This concept is crucial since it helps determine the value of the electric field (\( E \)), which is a vector quantity.
In our problem, the test charge (\( q_0 \)) is essential for measuring the resulting force (\( F \)). A crucial aspect of using a test charge is to assume it is small enough so that it does not alter the electric field it is measuring. This assumption ensures that the measurement of the electric field remains accurate.
The path of the test charge gives insight into the direction of the electric field. Electric fields are depicted as imaginary lines that indicate force direction, moving away from positive charges and toward negative charges. Thus, when a test charge is placed in an electric field, we analyze its interaction with the field to understand the field's impact and properties.

Force on Charge

The force on a charge in an electric field is essentially the interaction that a test charge experiences. In equation form, it is\( F = q_0 \times E \)where \( F \) is the force exerted, \( q_0 \) is the magnitude of the charge, and \( E \) is the electric field.
In this exercise, after placing a test charge (\( q_0 \)) at point (\( P \)), the force (\( F \)) acting on it helps us understand the actual strength and direction of the electric field. Calculating this force enables us to measure the electric field using the formula \( E = \frac{F}{q_0} \).
This understanding is grounded in the definition of electric fields, which states that the field is the force per unit charge. The relationship specifies that the region around a charged object where a force would be exerted on other charges present in that space.