Ch 25: The Electric Potential

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 25: The Electric Potential

Problem 39

Chapter 25, Problem 39

A +3.0 nC charge is at x=0 cm and a −1.0 nC charge is at x=4 cm. At what point or points on the x-axis is the electric potential zero?

Verified step by step guidance

Understand the problem: The electric potential at a point due to a charge is given by the formula \( V = \frac{kq}{r} \), where \( k \) is Coulomb's constant, \( q \) is the charge, and \( r \) is the distance from the charge to the point. The goal is to find the point(s) on the x-axis where the total electric potential from both charges is zero.

Set up the equation for the total electric potential: The total potential at a point \( x \) on the x-axis is the sum of the potentials due to the two charges. Let the distance from the +3.0 nC charge to the point be \( r_1 \) and the distance from the −1.0 nC charge to the point be \( r_2 \). The equation becomes \( \frac{k(3.0 \times 10^{-9})}{r_1} + \frac{k(-1.0 \times 10^{-9})}{r_2} = 0 \).

Express the distances \( r_1 \) and \( r_2 \) in terms of \( x \): Since the +3.0 nC charge is at \( x = 0 \) cm and the −1.0 nC charge is at \( x = 4 \) cm, \( r_1 = |x| \) and \( r_2 = |x - 4| \). Substitute these into the equation: \( \frac{k(3.0 \times 10^{-9})}{|x|} + \frac{k(-1.0 \times 10^{-9})}{|x - 4|} = 0 \).

Simplify the equation: Cancel out \( k \) (since it is a common factor) and rewrite the equation as \( \frac{3.0 \times 10^{-9}}{|x|} = \frac{1.0 \times 10^{-9}}{|x - 4|} \). Simplify further to \( \frac{3}{|x|} = \frac{1}{|x - 4|} \).

Solve for \( x \): Cross-multiply to get \( 3|x - 4| = |x| \). This equation can be solved by considering the different cases for \( x \) (e.g., \( x < 0 \), \( 0 < x < 4 \), and \( x > 4 \)) to find the point(s) where the electric potential is zero. Analyze each case to determine the valid solution(s).

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Key Concepts

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

Electric Potential

Electric potential is the amount of electric potential energy per unit charge at a point in an electric field. It is a scalar quantity measured in volts (V) and indicates the work done to move a unit positive charge from a reference point to a specific point in the field without any acceleration.

Superposition Principle

The superposition principle states that the total electric potential at a point due to multiple charges is the algebraic sum of the potentials due to each charge individually. This principle allows us to analyze complex charge configurations by considering the contributions from each charge separately.

Distance and Charge Relationship

The electric potential created by a point charge decreases with distance from the charge. Specifically, the potential (V) due to a point charge (Q) at a distance (r) is given by V = kQ/r, where k is Coulomb's constant. This relationship is crucial for determining where the total potential is zero, as it helps identify the influence of each charge at various points along the x-axis.

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Related Practice

Textbook Question

A −10.0 nC point charge and a +20.0 nC point charge are 15.0 cm apart on the x-axis. What is the electric potential at the point on the x-axis where the electric field is zero?

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Textbook Question

A −3.0 nC charge is on the x-axis at x=−9 cm and a +4.0 nC charge is on the x-axis at x=16 cm. At what point or points on the y-axis is the electric potential zero?

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Textbook Question

Two small metal cubes with masses 2.0 g and 4.0 g are tied together by a 5.0-cm-long massless string and are at rest on a frictionless surface. Each is charged to +2.0 μC. What is the tension in the string?

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Textbook Question

Two point charges qₐ and qᵦ are located on the x-axis at x=a and x=b. FIGURE EX25.36 is a graph of V, the electric potential. Draw a graph of Eₓ, the x-component of the electric field, as a function of x.

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Textbook Question

Two point charges 2.0 cm apart have an electric potential energy −180 μJ. The total charge is 30 nC. What are the two charges?

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Textbook Question

The four 1.0 g spheres shown in FIGURE P25.42 are released simultaneously and allowed to move away from each other. What is the speed of each sphere when they are very far apart?

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