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Chapter 15: Problem 13

Two charges are brought together until they are \(100 \mathrm{~cm}\) apart, causing the electric force between them to increase by a factor of exactly \(5 .\) What was their initial separation distance?

Short Answer

Expert verified
The initial separation distance was approximately 223.6 cm.

Step by step solution

01

Understanding the Problem

We are given that two charges are initially separated by a distance that we need to find. When they are brought to a distance of \(100 \text{ cm}\), the force between them increases by a factor of 5. We need to use the inverse square law to determine their initial separation.
02

Applying Coulomb's Law

Coulomb's Law states that the electric force \(F\) between two charges is given by \(F = k \frac{q_1 q_2}{r^2}\), where \(k\) is Coulomb's constant, \(q_1\) and \(q_2\) are the charges, and \(r\) is the distance between them. We are given \(r_2 = 100 \text{ cm} = 1 \text{ m}\).
03

Relating Initial and Final Forces

Let \(F_1\) be the initial force when the charges are \(r_1\) cm apart, and \(F_2\) be the force when they are 100 cm apart. Given \(F_2 = 5F_1\), we apply the force formula for both: \(F_1 = k \frac{q_1 q_2}{r_1^2}\) and \(F_2 = k \frac{q_1 q_2}{(1)^2}\).
04

Setting Up the Equation

Since \(F_2 = 5F_1\), substitute the force formulas: \(k \frac{q_1 q_2}{1} = 5 \left(k \frac{q_1 q_2}{r_1^2}\right)\). Simplifying, we get \(\frac{1}{1} = \frac{5}{r_1^2}\).
05

Solving for Initial Distance

From \(\frac{1}{1} = \frac{5}{r_1^2}\), solve for \(r_1^2\): \(r_1^2 = 5\). Thus, \(r_1 = \sqrt{5} \text{ m}\) or \(r_1 = \sqrt{5} \times 100 \text{ cm} \approx 223.6 \text{ cm}\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Coulomb's Law
Coulomb's Law is crucial for understanding how electrical forces operate between charged objects. This law provides a formula to calculate the strength of the electrical force. The formula is:
  • \( F = k \frac{q_1 q_2}{r^2} \)
Here, \( F \) represents the electric force, \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the magnitudes of the charges, and \( r \) is the distance between the two charges.

A few important points about Coulomb's Law:
  • The force is attractive if the charges are of opposite signs and repulsive if they have the same sign.
  • The stronger the charges, the stronger the force between them.
  • The effect of the force diminishes with increasing distance.
Inverse Square Law
The inverse square law plays a significant role in Coulomb's law. It dictates how the strength of electric force decreases with increasing distance between charges. According to this law, the force is inversely proportional to the square of the distance:
  • \( F \propto \frac{1}{r^2} \)

This means that if you double the distance \( r \), the force \( F \) becomes one-fourth as strong. Conversely, halving the distance means the force increases to four times its original value.

Understanding this relationship helps explain why bringing charges closer intensifies their interaction and vice versa.
Distance Between Charges
The distance between charges, denoted by \( r \), directly influences the magnitude of the electrical force acting between them. As per the inverse square law, a small change in the distance can lead to a substantial change in force.

In the example exercise, two charges are initially separated by an unknown distance. Upon reducing this distance to 100 cm, the force increases by a factor of 5, showcasing the distance's critical impact.
  • Shorter distances increase the force between charges.
  • Larger distances decrease the force.
Understanding the calculation and effect of distance helps in the accurate application of electric force in practical scenarios.
Electric Charges Interaction
Electric charges can be positive or negative, and the way they interact depends heavily on their types and magnitudes. This interaction can be either attractive or repulsive.

Key elements of charge interaction include:
  • Like charges repel each other, whereas opposite charges attract.
  • The greater the charge, the stronger the interaction force.
  • Closer proximity increases the force, amplifying the interaction.
These principles govern the behavior of all electric charge interactions, ranging from static electricity phenomena to complex electromagnetic systems. The initial distance in the exercise influenced the way these interactions played out as depicted by the increase in force when the distance was altered.

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Most popular questions from this chapter

Two charges, \(q_{1}\) and \(q_{2},\) are located at the origin and at \((0.50 \mathrm{~m}, 0),\) respectively. Where on the \(x\) -axis must a third charge, \(q_{3},\) of arbitrary sign be placed to be in electrostatic equilibrium if (a) \(q_{1}\) and \(q_{2}\) are like charges of equal magnitude, (b) \(q_{1}\) and \(q_{2}\) are unlike charges of equal magnitude, and (c) \(q_{1}=+3.0 \mu \mathrm{C}\) and \(q_{2}=-7.0 \mu \mathrm{C} ?\)

In walking across a carpet, you acquire a net negative charge of \(50 \mu \mathrm{C}\). How many excess electrons do you have?

Two identical point charges are a fixed distance apart. By what factor would the magnitude of the electric force between them change if (a) one of their charges were doubled and the other were halved, (b) both their charges were halved, and (c) one charge were halved and the other were left unchanged?

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?

Two negative point charges are separated by \(10.0 \mathrm{~cm}\) and feel a mutual repulsive force of \(3.15 \mu \mathrm{N}\). The charge of one is three times that of the other. (a) How much charge does each have? (b) What would be the force if the total charge were instead equally distributed on both point charges?
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