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 41

Chapter 25, Problem 41

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?

Verified step by step guidance

Step 1: Recall the formula for 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. The total electric potential at a point is the sum of the potentials due to all charges.

Step 2: Identify the coordinates of the charges and the point of interest. The \( -3.0 \, \text{nC} \) charge is at \( (-9, 0) \) cm, and the \( +4.0 \, \text{nC} \) charge is at \( (16, 0) \) cm. The point of interest lies on the y-axis, so its coordinates are \( (0, y) \).

Step 3: Write the expression for the distance from each charge to the point on the y-axis. For the \( -3.0 \, \text{nC} \) charge, the distance is \( r_1 = \sqrt{(-9)^2 + y^2} \). For the \( +4.0 \, \text{nC} \) charge, the distance is \( r_2 = \sqrt{16^2 + y^2} \).

Step 4: Set up the equation for the electric potential to be zero at the point on the y-axis. The total potential is \( V = \frac{k(-3.0 \times 10^{-9})}{r_1} + \frac{k(4.0 \times 10^{-9})}{r_2} \). For \( V = 0 \), this simplifies to \( \frac{-3.0}{r_1} + \frac{4.0}{r_2} = 0 \).

Step 5: Solve for \( y \) by substituting \( r_1 \) and \( r_2 \) into the equation. This gives \( \frac{-3.0}{\sqrt{(-9)^2 + y^2}} + \frac{4.0}{\sqrt{16^2 + y^2}} = 0 \). Rearrange and solve for \( y \) algebraically to find the point(s) on the y-axis where the electric potential is zero.

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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, measured in volts, is the amount of electric potential energy per unit charge at a point in space. It is a scalar quantity that represents the work done to move a unit positive charge from infinity to that point. The electric potential due to a point charge decreases with distance from the charge, and the total potential at a point is the algebraic sum of the potentials due to all charges present.

Superposition Principle

The superposition principle states that the total electric potential at a point due to multiple charges is the sum of the potentials due to each charge considered independently. This principle allows us to analyze complex charge configurations by calculating the potential from each charge separately and then combining the results. It is essential for solving problems involving multiple charges, such as finding points where the total potential is zero.

Coordinate System and Symmetry

In this problem, the coordinate system is crucial for determining the positions of the charges and the points on the y-axis where the electric potential is zero. The symmetry of the charge distribution can simplify the analysis, as the electric potential will be influenced by the relative distances from the charges. Understanding how to set up the coordinate system and recognizing symmetrical properties can help identify potential points more efficiently.

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