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Ch 23: The Electric Field
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 23: The Electric FieldProblema 10
Capitolo 23, Problema 10

A small glass bead charged to +6.0 nC is in the plane that bisects a thin, uniformly charged, 10-cm-long glass rod and is 4.0 cm from the rod's center. The bead is repelled from the rod with a force of 840 μN. What is the total charge on the rod?

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Identify the relevant physical principle: The force between the charged bead and the charged rod is due to the electric field produced by the rod. The force on the bead is given by \( F = q \cdot E \), where \( F \) is the force, \( q \) is the charge of the bead, and \( E \) is the electric field at the bead's location.
Determine the electric field produced by the rod: Since the rod is uniformly charged and the bead is located in the plane that bisects the rod, the electric field at the bead's location can be calculated using the formula for the electric field due to a uniformly charged finite line of charge. The formula is \( E = \frac{k \cdot \lambda \cdot L}{r \cdot \sqrt{r^2 + (L/2)^2}} \), where \( k \) is Coulomb's constant, \( \lambda \) is the linear charge density, \( L \) is the length of the rod, and \( r \) is the perpendicular distance from the rod to the bead.
Relate the linear charge density \( \lambda \) to the total charge \( Q \): The linear charge density is defined as \( \lambda = \frac{Q}{L} \), where \( Q \) is the total charge on the rod and \( L \) is the length of the rod. Substitute \( \lambda \) into the electric field formula to express \( E \) in terms of \( Q \).
Use the force equation to solve for \( Q \): Substitute the expression for \( E \) into \( F = q \cdot E \). This gives \( F = q \cdot \frac{k \cdot Q}{L \cdot r \cdot \sqrt{r^2 + (L/2)^2}} \). Rearrange this equation to solve for \( Q \): \( Q = \frac{F \cdot L \cdot r \cdot \sqrt{r^2 + (L/2)^2}}{k \cdot q} \).
Substitute the known values into the equation: Use \( F = 840 \ \mu\text{N} = 840 \times 10^{-6} \ \text{N} \), \( q = 6.0 \ \text{nC} = 6.0 \times 10^{-9} \ \text{C} \), \( L = 10 \ \text{cm} = 0.10 \ \text{m} \), \( r = 4.0 \ \text{cm} = 0.04 \ \text{m} \), and \( k = 8.99 \times 10^9 \ \text{N·m}^2/\text{C}^2 \). Substitute these values into the formula for \( Q \) to calculate the total charge on the rod.

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Concetti chiave

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Coulomb's Law

Coulomb's Law describes the electrostatic force between two charged objects. It states that the force (F) is directly proportional to the product of the magnitudes of the charges (q1 and q2) and inversely proportional to the square of the distance (r) between them. Mathematically, it is expressed as F = k * (|q1 * q2| / r^2), where k is Coulomb's constant. This law is fundamental for calculating the forces between charged particles.
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Electric Field

An electric field is a region around a charged object where other charges experience a force. It is defined as the force per unit charge experienced by a positive test charge placed in the field. The electric field (E) due to a point charge can be calculated using E = k * (|q| / r^2), where q is the charge creating the field and r is the distance from the charge. Understanding electric fields is crucial for analyzing the interactions between charged objects.
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Intro to Electric Fields

Superposition Principle

The superposition principle states that the total electric field or force acting on a charge due to multiple other charges is the vector sum of the individual fields or forces produced by each charge. This principle allows us to analyze complex charge configurations by breaking them down into simpler components. In the context of the problem, it helps in determining the total charge on the rod by considering the contributions from the bead and the rod's charge.
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