Problem 21 - College Physics

Question

If the distance from a charge is doubled, is the magnitude of the electric field (1) increased, (2) decreased, or (3) the same compared to the initial value? (b) If the original electric field due to a charge is \(1.0 \times 10^{-4} \mathrm{~N} / \mathrm{C},\) what is the magnitude of the new electric field at twice the distance from the charge?

Step-by-Step Solution

Answer

The electric field decreases by a factor of 4 to \(2.5 \times 10^{-5}\) N/C when the distance is doubled.

Step 1: Understand the Relationship

The electric field (E) due to a point charge is given by \( E = \frac{k \cdot |q|}{r^2} \), where \( k \) is a constant (Coulomb's constant), \( q \) is the charge, and \( r \) is the distance from the charge. As the distance \( r \) increases, the electric field \( E \) decreases.

Step 2: Determine the Effect of Doubling the Distance

Doubling the distance means replacing \( r \) with \( 2r \). The new electric field \( E' \) becomes \( E' = \frac{k \cdot |q|}{(2r)^2} = \frac{k \cdot |q|}{4r^2} = \frac{1}{4}E \). Thus, doubling the distance decreases the electric field by a factor of 4.

Step 3: Calculate the New Electric Field

Given the original electric field \( E = 1.0 \times 10^{-4} \mathrm{~N}/\mathrm{C} \), the new electric field \( E' \) at twice the distance is \( E' = \frac{1}{4} \times 1.0 \times 10^{-4} = 2.5 \times 10^{-5} \mathrm{~N}/\mathrm{C} \).

Step 4: Conclude the Observations

When the distance from the charge is doubled, the electric field decreases by a factor of 4, from \( 1.0 \times 10^{-4} \mathrm{~N}/\mathrm{C} \) to \( 2.5 \times 10^{-5} \mathrm{~N}/\mathrm{C} \). Thus, the magnitude has decreased.

Key Concepts

Coulomb's Law

Coulomb's Law is a fundamental principle in electromagnetism that describes the force between two point charges. It states that the electric force (\( F \)) between two charges is directly proportional to the product of the magnitudes of the charges (\( |q_1 q_2| \)) and inversely proportional to the square of the distance (\( r \)) between them. This can be mathematically expressed as:
\[ F = k \frac{|q_1 q_2|}{r^2} \]
Where \( k \) is Coulomb's constant, approximately 8.99 x 10^9 N m²/C². This law provides the foundation for understanding electric fields and how point charges interact. When analyzing electric fields created by a point charge, we use a closely related formula derived from Coulomb's Law, which calculates the electric field (\( E \)) as:
\[ E = \frac{k \cdot |q|}{r^2} \]
This calculation helps determine how an electric field diminishes with distance, a key part of the original exercise.

Point Charge

In physics, a point charge is an idealized model of a charged particle. It is considered to be infinitely small, with all of its electric charge concentrated at a single point in space. This simplification allows easier calculation of electric fields and forces between charges.

• It helps to simplify complex problems by treating real objects with charge as if they were infinitely small.
• Point charges are used to illustrate theoretical concepts and serve as the foundational elements in electrical interactions.
• When calculating the electric field around a point charge, the formula used is \( E = \frac{k \cdot |q|}{r^2} \), where \( r \) is the distance from the point charge.

Understanding the behavior of point charges is essential for determining how electric fields vary and how multiple charges interact with each other. This application was central to solving the problem about how distance affects the electric field.

Inverse Square Law

The Inverse Square Law is a crucial principle in physics that relates to forces such as gravity and electromagnetism. This law states that a given physical quantity or force is inversely proportional to the square of the distance from the source. For electric fields, we see this in how the field intensity decreases rapidly as distance increases.

• This principle is embedded in the equation \( E = \frac{k \cdot |q|}{r^2} \), emphasizing how increasing the distance drastically reduces field strength.
• The inverse square relationship is why doubling the distance from a point charge decreases the electric field's strength by a factor of four, as seen in the problem.

By understanding the inverse square nature of force laws, students can predict how changes in distance will affect forces and fields in a wide range of physical contexts. This relationship helps explain phenomena over large distances and is essential for grasping electric field behaviors.