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Ch. 21 - Electric Charge and Electric Field
Giancoli Douglas - Physics for Scientists and Engineers 5th edition
Giancoli Douglas5th editionPhysics for Scientists and EngineersISBN: 9780137488179Not the one you use?Change textbook
All textbooksGiancoli Douglas 5th editionCh. 21 - Electric Charge and Electric FieldProblem 11
Chapter 21, Problem 11

Two positive point charges are a fixed distance apart. The sum of their charges is Qₜ. What charge must each have in order to
(a) maximize the electric force between them, and
(b) minimize it?

Verified step by step guidance
1
Step 1: Recall Coulomb's Law, which states that the electric force between two point charges is given by \( F = \frac{k \cdot q_1 \cdot q_2}{r^2} \), where \( k \) is Coulomb's constant, \( q_1 \) and \( q_2 \) are the charges, and \( r \) is the distance between them.
Step 2: Express the relationship between the charges \( q_1 \) and \( q_2 \) using the given total charge \( Q_t \). Since the sum of the charges is \( Q_t \), we can write \( q_1 + q_2 = Q_t \). Let \( q_1 = x \) and \( q_2 = Q_t - x \).
Step 3: Substitute \( q_1 = x \) and \( q_2 = Q_t - x \) into Coulomb's Law to express the force as a function of \( x \): \( F(x) = \frac{k \cdot x \cdot (Q_t - x)}{r^2} \). Simplify this to \( F(x) = \frac{k \cdot Q_t \cdot x - k \cdot x^2}{r^2} \).
Step 4: To maximize the force, find the value of \( x \) that maximizes \( F(x) \). This requires taking the derivative of \( F(x) \) with respect to \( x \), setting it equal to zero, and solving for \( x \). The derivative is \( \frac{dF}{dx} = \frac{k \cdot Q_t}{r^2} - \frac{2k \cdot x}{r^2} \). Set \( \frac{dF}{dx} = 0 \) and solve for \( x \).
Step 5: To minimize the force, consider the extreme cases where one charge approaches zero (\( q_1 \to 0 \)) and the other approaches \( Q_t \) (\( q_2 \to Q_t \)), or vice versa. Substitute these values into the force equation \( F = \frac{k \cdot q_1 \cdot q_2}{r^2} \) to confirm that the force approaches zero in these cases.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Coulomb's Law

Coulomb's Law describes the electrostatic force between two point charges. It states that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This law is fundamental in understanding how charges interact, as it quantifies the strength and direction of the force based on charge values and separation distance.
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Coulomb's Law

Charge Distribution

Charge distribution refers to how total charge Qₜ is allocated between two point charges. To maximize the electric force, both charges should be equal, meaning each charge would be Qₜ/2. Conversely, to minimize the force, one charge should be zero, leaving the other with the entire charge Qₜ, as the force depends on the product of the charges.
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Probability Distribution Graph

Electric Force Maximization and Minimization

Maximizing and minimizing electric force involves understanding how the configuration of charges affects the resultant force. The maximum force occurs when both charges are equal, maximizing their product. In contrast, the minimum force occurs when one charge is zero, resulting in no interaction, illustrating how charge values directly influence the strength of the electric force.
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Work due to Electric Force
Related Practice
Textbook Question

Draw, approximately, the electric field lines emanating from a uniformly charged straight wire whose length ℓ is not great. The spacing between lines near the wire should be much less than ℓ. [Hint: Also consider points very far from the wire up to 4ℓ \.]

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Textbook Question

An electron moving to the right at 7.5 x 10⁵ m/s enters a uniform electric field parallel to its direction of motion. If the electron is to be brought to rest in the space of 5.0 cm,

(a) what direction is required for the electric field, and

(b) what is the strength of the field?

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Textbook Question

You are given two unknown point charges, Q₁ and Q₂. At a point on the line joining them, one-third of the way from Q₁ to Q₂, the electric field is zero (Fig. 21–64). What is the ratio Q₁/Q₂?