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Chapter 19: Problem 18

A charge of \(+125 \mu C\) is fixed at the center of a square that is \(0.64 \mathrm{m}\) on a side. How much work is done by the electric force as a charge of \(+7.0 \mu \mathrm{C}\) is moved from one corner of the square to any other empty corner? Explain.

Short Answer

Expert verified
No work is done as the charge moves between equidistant corners in a symmetric electric field.

Step by step solution

01

Identify Initial and Final Positions

Let's start by identifying the initial and final positions of the charge. The charge of \(+7.0 \,\mu \mathrm{C}\) moves from one corner of the square to another. The square is \(0.64 \, m\) on a side. We will calculate the distances involved in moving a charge between corners.
02

Calculate Distance Between Diagonal Corners

To find the work done by the electric force, we need to find the distance moved diagonally across the square. Using the Pythagorean theorem for a square of side length \(0.64 \, \mathrm{m}\), the diagonal distance is \(\sqrt{(0.64)^2 + (0.64)^2} \). Calculating this gives: \(d_{\text{diagonal}} = 0.64 \sqrt{2} \, \mathrm{m} \).
03

Understand Electric Field Symmetry

Since both the initial and final positions are equidistant from the center of the square, the work done by the electric force in moving the charge across this diagonal must account for this symmetry. As a result, there is no net change in potential energy because the path integral across a symmetrical field with no potential difference is zero.
04

Calculate Work Using Electric Potential Difference

The work done by an electric field when moving a charge is given by \(W = q \Delta V\), where \(\Delta V\) is the potential difference between initial and final positions. Given the potential at each equidistant position from the central charge is the same, the potential difference \(\Delta V = 0\) implies that \(W = (7.0 \, \mu \mathrm{C})(0) = 0\). Thus, no work is done.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Electric Potential
Electric potential, often denoted by the symbol \( V \), is a measure of the potential energy per unit charge at a point in an electric field. It's analogous to how gravitational potential determines how much potential energy an object might have because of its position in a gravitational field.
Key Points about Electric Potential:
  • The electric potential at a point in space is the amount of work needed to bring a unit positive charge from an arbitrarily chosen reference point to this point.
  • It is often measured in volts (V), where \(1 \, \text{volt} = 1 \, \text{joule per coulomb}\).
  • Electric potential can vary from one location to another in an electric field. However, when two points have the same electric potential, no net work is required to move a charge between these locations, as the potential difference \(\Delta V\) is zero.
In the context of the exercise, since the charge is moved between corners of a square and remains equidistant from the central charge, the electric potential at these corners is equal. Thus, moving a charge between these corners requires no work.
Work Done by Electric Force
The concept of work done by an electric force is similar to work done by other forces. Work is generally defined as the energy transferred by a force acting over a distance. However, in an electric field:
  • The electric force does work when moving a charge through an electric potential difference.
  • The work \( W \) done is calculated using the formula: \( W = q \Delta V \), where \( q \) is the charge and \( \Delta V \) is the potential difference between two points.
  • When \( \Delta V = 0 \), no work is done because there is no change in potential energy. This scenario occurs when moving between points at the same electric potential.

For the exercise, because all corners are equidistant from the center where the charge is fixed, there is no potential difference between the corners, leading to zero work done by the electric force.
Symmetry in Electric Fields
Symmetry plays a crucial role in simplifying the analysis of electric fields and potentials. When we talk about symmetry in electric fields:
  • It refers to situations where the electric field distribution allows simplifications, typically due to geometric arrangements.
  • For instance, in a symmetric set-up, such as a square with a charge at the center, each corner of the square is equidistant to the central charge, leading to symmetrical potential distribution.
  • This symmetry implies that the electric potential at all corners of the square is the same if they are equally distant from the charge.
In the given exercise, the symmetry of the square ensures that the potential at each corner remains constant throughout the movement, leading to no change in potential energy and thus no work is required to move the charge between corners.

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Most popular questions from this chapter

A positive charge \(q_{1}\) is located \(3.00 \mathrm{m}\) to the left of a negative charge \(q_{2} .\) The charges have different magnitudes. On the line through the charges, the net electric field is zero at a spot \(1.00 \mathrm{m}\) to the right of the negative charge. On this line there are also two spots where the total electric potential is zero. Locate these two spots relative to the negative charge.

An electron and a proton are initially very far apart (effectively an infinite distance apart). They are then brought together to form a hydrogen atom, in which the electron orbits the proton at an average distance of \(5.29 \times 10^{-11} \mathrm{m} .\) What is \(\mathrm{EPE}_{\text {final }}-\mathrm{EPE}_{\text {initial }},\) which is the change in the electric potential energy?

Particle 1 has a mass of \(m_{1}=3.6 \times 10^{-6} \mathrm{kg},\) while particle 2 has a mass of \(m_{2}=6.2 \times 10^{-6} \mathrm{kg} .\) Each has the same electric charge. These particles are initially held at rest, and the two- particle system has an initial electric potential energy of 0.150 J. Suddenly, the particles are released and fly apart because of the repulsive electric force that acts (a) Two particles have different masses, but the same electrical charge \(q\) They are initially at rest. (b) At the instant following the release of the particles, they are flying apart due to the mutual force of electric repulsion. on each one (see the figure). The effects of the gravitational force are negligible, and no other forces act on the particles. Concepts: (i) What types of energy does the two-particle system have initially? (ii) What types of energy does the two-particle system have at the instant illustrated in part \(b\) of the drawing? (iii) Does the principle of conservation of energy apply to this problem? Explain. (iv) Does the conservation of linear momentum apply to the two particles as they fly apart? Explain. Calculations: At one instant following the release, the speed of particle 1 is measured to be \(v_{1}=170 \mathrm{m} / \mathrm{s} .\) What is the electric potential energy at this instant?

Two equipotential surfaces surround a \(+1.50 \times 10^{-8} \mathrm{C}\) point charge. How far is the \(190-\mathrm{V}\) surface from the \(75.0-\mathrm{V}\) surface?

At a distance of \(1.60 \mathrm{m}\) from a point charge of \(+2.00 \mu \mathrm{C}\), there is an equipotential surface. At greater distances there are additional equipotential surfaces. The potential difference between any two successive surfaces is \(1.00 \times 10^{3} \mathrm{V} .\) Starting at a distance of \(1.60 \mathrm{m}\) and moving radially outward, how many of the additional equipotential surfaces are crossed by the time the electric field has shrunk to one-half of its initial value? Do not include the starting surface.
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