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 35

Chapter 25, Problem 35

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?

Verified step by step guidance

Identify the symmetry of the problem: The rod is divided into two halves, one positively charged (+Q) and the other negatively charged (-Q). The point of interest lies along the axis of symmetry of the rod, simplifying the calculation of the electric potential.

Recall the formula for the electric potential due to a point charge: \( V = \frac{kQ}{r} \), where \( k \) is Coulomb's constant, \( Q \) is the charge, and \( r \) is the distance from the charge to the point of interest. For a continuous charge distribution, the potential is calculated by integrating over the charge distribution.

Set up the integral for the electric potential: Divide the rod into infinitesimal charge elements \( dq \). For each element, the potential contribution at the point is \( dV = \frac{k dq}{r} \), where \( r \) is the distance from the charge element to the point. Integrate this expression over the length of each half of the rod.

Express \( dq \) in terms of the linear charge density \( \lambda \): The linear charge density is \( \lambda = \frac{Q}{L} \), where \( L \) is the length of each half of the rod. Thus, \( dq = \lambda dx \), where \( dx \) is an infinitesimal length element of the rod.

Perform the integration for each half of the rod: For the positively charged half, integrate \( \frac{k \lambda dx}{r} \) over its length. For the negatively charged half, do the same but with a negative charge density. Add the contributions from both halves to find the total electric potential at the point.

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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, often referred to as voltage, is the amount of electric potential energy per unit charge at a specific 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 the specified point without any acceleration.

Superposition Principle

The superposition principle states that the total electric potential at a point due to multiple charge distributions is the algebraic sum of the potentials due to each charge considered independently. This principle allows us to calculate the net electric potential by adding the contributions from each charged object, taking into account their respective distances from the point of interest.

Uniform Charge Distribution

Uniform charge distribution refers to a scenario where charge is spread evenly over a given length, area, or volume. In the context of the rod described, each half is uniformly charged to ±Q, meaning that the charge density is constant along the length of each half. This uniformity simplifies calculations of electric fields and potentials, as the effects of each segment can be treated uniformly.

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Related Practice

Textbook Question

Two small charged spheres are 5.0 cm apart. One is charged to +25 nC, the other to −15 nC. A proton is released from rest halfway between the spheres. What is the proton's speed after it has moved 1.0 cm?

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Textbook Question

What is the electric potential at the point indicated with the dot in FIGURE EX25.31?

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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

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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Textbook Question

Four small spheres, each charged to +15 nC, form a square 2.0 cm on each side. From far away, a proton is shot toward the square along a line perpendicular to the square and passing through its center. What minimum initial speed does the proton need to pass through the square of charges?

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