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Chapter 20: Problem 67

Make a qualitative sketch of the electric field lines produced by three charges, \(+q,+q\), and \(-q\), positioned at the corners of an equilateral triangle with sides of length \(d\).

Short Answer

Expert verified
Field lines begin at +q, end at -q, and show repulsion between +q charges.

Step by step solution

01

Identify Charge Configuration

We have three charges: two positive charges, each denoted as \(+q\), and one negative charge, denoted as \(-q\). These are placed at the corners of an equilateral triangle with side length \(d\). Let's label the corners as A, B, and C where A and B have \(+q\) while C has \(-q\).
02

Understand Field Line Properties

Electric field lines start from positive charges and end on negative charges. The number of lines originating or terminating on a charge is proportional to the magnitude of the charge. For equal magnitude charges, lines emanate equally in all directions.
03

Sketch Field Lines from Positive Charges

Draw several lines originating from both \(+q\) charges at corners A and B. Since they are like charges, some lines will point away from each other. The lines from A and B will bend towards the \(-q\) charge at C, since field lines aim to terminate on negative charges.
04

Sketch Field Lines to Negative Charge

Draw lines from the charges at A and B converging towards the \(-q\) charge at C, demonstrating the attraction between opposite charges. Due to symmetry in this triangular arrangement, lines from both A and B should connect to C.
05

Complete Field Line Patterns

Field lines between like charges at A and B will demonstrate repulsion, somewhat ballooning out before meeting the field lines from C. Field lines between A and C, and B and C, will not cross and are directed along the shortest path between unlike charges.

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Key Concepts

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

Charge Configuration
Understanding the charge configuration is the first step in sketching electric field lines. In our given problem, we have three charges forming the vertices of an equilateral triangle. This specific configuration consists of two positive charges, both labeled as \(+q\), and one negative charge, labeled as \(-q\). Here are some key points to remember when considering charge configurations:
  • Identifying the type and number of charges is crucial, as it influences the direction and origin of field lines.
  • In this configuration, the charges create a symmetric shape, which simplifies determining the electric field directions.
  • The placement of these charges determines how the field lines intersect and attract or repel each other across the triangle.
Recognizing these basic attributes helps set a solid foundation for visualizing electric fields effectively.
Electric Field Properties
Electric field lines provide a visual way to understand electric fields around charges. The properties of electric field lines follow specific rules that you need to keep in mind:
  • Field lines originate from positive charges and terminate on negative charges.
  • The number of lines indicates the strength of the charge; equal magnitude charges have an equal number of lines.
  • The lines never intersect, as each point in the space around the charge has a unique direction.
In the current charge configuration, you will see lines starting from the positive charges at locations A and B. These lines will aim towards the negative charge at location C. This behavior illustrates both the directional properties and the tendency of field lines to connect opposite charges.
Field Line Sketching
Sketching field lines can seem complex, but by using the properties of electric fields, it becomes more manageable. Here鈥檚 how you can sketch electric field lines for our scenario:
  • Begin by drawing lines away from the positive charge at corner A and B. These lines need to curve away from each other slightly due to their mutual repulsion.
  • Next, guide these lines towards the \(-q\) charge at corner C, illustrating the attraction between opposite charges.
  • Due to the symmetry in the equilateral triangle, ensure that equivalent lines are drawn from both positive charges leading to the negative charge.
The final sketch for the triangle should show a balanced pattern of lines, where lines spread out from the positive charges and converge towards the negative charge. This accurately reflects the concept of opposing type attractions and like type repulsions.
Charge Interactions
Understanding how charges interact is essential for field line sketching. Let's break down their interactions in our example:
  • Positive charges at corners A and B repel each other, creating a ballooning effect of the field lines starting from them.
  • Both positive charges are attracted towards the negative charge at corner C, drawing field lines inward toward C.
  • The repulsion and attraction balance should maintain a symmetric and predictable pattern around the triangle.
Always remember that field lines can help in visualizing these interactions. They give insight into how the fields influence each other and move toward or away from each other, creating a dynamic yet stable field configuration.

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Most popular questions from this chapter

A capacitor stores \(2.7 \times 10^{-5} \mathrm{C}\) of charge when it is connected to a \(9.0-\mathrm{V}\) battery. What is the capacitance of the capacitor?

In one region of space the electric potential has a positive constant value. In another region of space the potential has a negative constant value. What can be said about the electric field within each of these two regions of space?

A small object with mass \(0.0150 \mathrm{~kg}\) and charge \(3.1 \mu \mathrm{C}\) hangs from the ceiling by a thread. A second small object, with a charge of \(4.2 \mu \mathrm{C}\), is placed \(1.2 \mathrm{~m}\) vertically below the first charge. Find (a) the electric field at the position of the upper charge due to the lower charge and (b) the tension in the thread.

Long, long ago, on a planet far, far away, a physics experiment was carried out. First, a \(0.250-\mathrm{kg}\) ball with zero net charge was dropped from rest from a height of \(1.00 \mathrm{~m}\). The ball landed \(0.552 \mathrm{~s}\) later. Next, the ball was given a net charge of \(7.75 \mu \mathrm{C}\) and dropped from rest from the same height. This time the ball fell for \(0.680 \mathrm{~s}\) before landing. What is the electric potential at a height of \(1.00 \mathrm{~m}\) above the ground on this planet, given that the electric potential at ground level is zero? (Air resistance can be ignored.)

A \(0.31-\mu \mathrm{F}\) capacitor is charged by a \(12-\mathrm{V}\) battery. If this capacitor is connected to an electric motor in a 16-kg go-kart, what speed can the go-kart acquire if it accelerates from rest? Assume \(100 \%\) efficiency for the motor and for the energy transfer from the capacitor to the motor.
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