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 37

Chapter 25, Problem 37

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?

Verified step by step guidance

Start by recalling the formula for electric potential energy between two point charges:

U = (k q

₁q₂)r, where

U

is the potential energy,

k

is Coulomb's constant (

8.99 × 10

⁹ N·m²/C²),

q

₁ and

q

₂ are the charges, and

r

is the distance between them.

Substitute the given values into the formula:

- 180 × 10

^-6 J = (8.99 × 10⁹ × q₁ × q₂)0.02. Rearrange the equation to solve for the product of the charges:

q

₁q₂ = (-180 × 10^-6 × 0.02)8.99 × 10⁹.

Next, use the fact that the total charge is given as

q

₁ + q₂ = 30 × 10^-9 C. Let

q

₁ = x and

q

₂ = 30 × 10^-9 - x.

Substitute

q

₁ and

q

₂ into the product equation:

x × (30 × 10

^-9 - x) = (-180 × 10^-6 × 0.02)8.99 × 10⁹. Expand and simplify the equation to form a quadratic equation in terms of

x

Solve the quadratic equation using the quadratic formula:

x = - b ± \sqrt{b}

² - 4ac2a, where

a

b

, and

c

are the coefficients of the quadratic equation. The two solutions will give the values of

q

₁ and

q

₂.

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Key Concepts

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

Electric Potential Energy

Electric potential energy is the energy a charged object possesses due to its position in an electric field. It is defined as the work done to move a charge from a reference point to a specific point in the field. The formula for the potential energy (U) between two point charges (q1 and q2) separated by a distance (r) is given by U = k * (q1 * q2) / r, where k is Coulomb's constant.

Coulomb's Law

Coulomb's Law describes the force between two point charges. It states that the magnitude of the electrostatic force (F) between two charges is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. Mathematically, it is expressed as F = k * (|q1 * q2|) / r², where k is Coulomb's constant. This law is fundamental in understanding the interactions between charged particles.

Charge Conservation

Charge conservation is a fundamental principle in physics stating that the total electric charge in an isolated system remains constant over time. In the context of the problem, this means that if the total charge of the two point charges is known, the individual charges can be determined by using the total charge and the relationship between potential energy and charge. This principle is crucial for solving problems involving multiple charges.

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Conservation of Charge

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 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?

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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

The two halves of the rod in FIGURE EX25.35 are uniformly charged to ±Q. What is the electric potential at the point indicated by the dot?

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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. What is the ratio |qₐ/qᵦ|?

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