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Chapter 21: Problem 21

A positive point charge \(q\) is placed on the \(+y\) -axis at \(y=a\) and a negative point clarge \(-q\) is placed on the \(-y\) -axis at \(y=-a\) . A negative point charge \(-Q\) is located at some point on the \(+x\) -axis, (a) In a free- body diagram, show the forces that act on the charge \(-Q\) . (b) Find the \(x\) - and \(y\) -components of the net force that the two charges \(q\) and \(-q\) exert on \(-Q .\) (Your answer should involve only \(k, q, Q, a\) and the coordinate \(x\) of the third charge. \()\) (c) What is the net force on the charge \(-Q\) when it is at the origin \((x=0)\) ? (d) Graph the \(y\) -component of the net force on the charge \(-Q\) as a function of \(x\) for values of \(x\) between \(-4 a\) and \(+4 a\) .

Short Answer

Expert verified
(a) Free-body diagram shows forces from \(+q\) and \(-q\). (b) Net force components: \(F_{net_x} = 0\), \(F_{net_y} = -2 \frac{k q Q a}{(x^2 + a^2)^{3/2}}\). (c) At origin, \(F_{net_y} = -2 \frac{k q Q}{a^2}\). (d) Graph \(F_{net_y}\) as symmetric, peaking near \(\pm a\).

Step by step solution

01

Understanding the Setup

We have three charges: a positive charge \(+q\) at \((0, a)\), a negative charge \(-q\) at \((0, -a)\), and a negative charge \(-Q\) placed at some point on the \(+x\)-axis, \((x, 0)\). We need to analyze the forces each charge exerts on \(-Q\).
02

Identify Forces in Free-Body Diagram

Draw a free-body diagram for the charge \(-Q\). The positive charge \(+q\) exerts an attractive force directed towards \(+q\), and the negative charge \(-q\) exerts a repulsive force directed away from \(-q\). These forces will have both \(x\) and \(y\) components.
03

Calculate Force from +q

The force exerted by \(+q\) on \(-Q\) is attractive. Using Coulomb's law, the magnitude of the force is \[ F_{+q} = \frac{k |q| |Q|}{(x^2 + a^2)}. \]The angle \(\theta\) between this force and the \(x\)-axis is given by \(\tan \theta = \frac{a}{x}\). So,\[ F_{+q_x} = F_{+q} \cdot \frac{x}{\sqrt{x^2+a^2}}, \quad F_{+q_y} = F_{+q} \cdot \frac{a}{\sqrt{x^2+a^2}}. \]
04

Calculate Force from -q

The force exerted by \(-q\) on \(-Q\) is repulsive. Similarly, the magnitude of the force is \[ F_{-q} = \frac{k |q| |Q|}{(x^2 + a^2)}. \]By symmetry, the angle \(\theta\) is the same as for \(+q\). So,\[ F_{-q_x} = -F_{-q} \cdot \frac{x}{\sqrt{x^2+a^2}}, \quad F_{-q_y} = -F_{-q} \cdot \frac{a}{\sqrt{x^2+a^2}}. \]
05

Find Net Force Components

The net force acting on \(-Q\) is the sum of the forces exerted by \(+q\) and \(-q\). So,\[ F_{net_x} = F_{+q_x} + F_{-q_x} = 0 \]because they are equal and opposite. For the \(y\)-component, we have\[ F_{net_y} = F_{+q_y} + F_{-q_y} = -2 \cdot \frac{k q Q a}{(x^2 + a^2)^{3/2}}. \]
06

Net Force at the Origin (c)

When \(-Q\) is at the origin, \(x = 0\).The \(y\)-component of the net force is\[ F_{net_y} = -2 \cdot \frac{k q Q a}{a^3} = -2 \cdot \frac{k q Q}{a^2}. \]The \(x\)-component remains zero.
07

Graphing the y-Component (d)

To graph the \(y\)-component of the force as a function of \(x\), plot \[ F_{net_y} = -2 \cdot \frac{k q Q a}{(x^2 + a^2)^{3/2}} \]for \(x\) values ranging from \(-4a\) to \(+4a\). This function is symmetric around the origin and shows maximum magnitude as \(x\) approaches \(卤a\).

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Key Concepts

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

Coulomb's Law
Coulomb's Law is a fundamental principle that helps us understand how charges interact. It states that the force between two point charges is directly proportional to the product of their magnitudes, inversely proportional to the square of the distance between them, and acts along the line joining the charges. The formula is given by:\(\[ F = \frac{k |q_1| |q_2|}{r^2} \]\)where \(F\) is the force between the charges, \(q_1\) and \(q_2\) are the magnitudes of the charges, \(r\) is the distance between the charges, and \(k\) is Coulomb's constant (\(8.99 \times 10^9 \, \text{Nm}^2/\text{C}^2\)).
  • This law helps calculate both the attraction and repulsion forces based on charge properties.

  • It applies to both static (charge at rest) and dynamic (charge in motion) electric field scenarios.
In problems involving multiple charges, such as in this exercise, Coulomb's Law is vital for determining the forces each charge exerts on another.
Free-Body Diagram
A free-body diagram is an essential tool in physics for visualizing the forces acting on an object. To construct a free-body diagram, follow these steps:
  • Identify the object of interest, in this case, the charge $-Q$.

  • Draw the object as a simple point or box.

  • Represent each force acting on the object with arrows pointing in the direction of the force, labeled with force names.
In our exercise, the forces in play are:
  • The attractive force from the positive charge $+q$ directed towards $+y$.

  • The repulsive force from the negative charge $-q$ also directed towards $-y$.
The diagram serves as a clear visual representation, helping us to break down the forces into components and solve for net effects.
Point Charge
Point charges are an idealized model where the charge is assumed to be concentrated at a single point in space. This simplification helps in calculating electric forces and fields without worrying about the object's size or shape. There are some key aspects to consider:
  • Point charges can be positive or negative, influencing whether they exert attractive or repulsive forces on other charges.

  • They are assumed to have no dimensions, which simplifies the application of Coulomb's Law and diagrammatic representations.
In our given setup:
  • The charge $+q$ is a point charge located at $(0, a)$, exerting an attractive force.

  • The charge $-q$ is at $(0, -a)$, generating a repulsive force.
Understanding the characteristics of point charges is crucial for accurately analyzing and calculating forces in electric fields.
Net Force Calculation
Calculating the net force is the final step that sums up all the forces acting on a charge to determine its resultant motion or equilibrium.Firstly, we dissect the forces into their components:
  • The \(x\)-components of the forces from \(+q\) and \(-q\) on \(-Q\) cancel each other out because they act in opposite directions and are equal in magnitude.
  • For the \(y\)-components, both act downwards towards the origin, so they add up, leading to a net force in the \(y\) direction.
The net force can be calculated using:\[ F_{net_y} = -2 \cdot \frac{k q Q a}{(x^2 + a^2)^{3/2}} \] When \(-Q\) is placed at the origin \((x = 0)\): \[ F_{net_y} = -2 \cdot \frac{k q Q}{a^2} \]By understanding how to break forces into components and add them together, net force calculations become a straightforward process that reveals the overall effect of multiple forces on a point charge.

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Most popular questions from this chapter

A charge of \(-6.50 \mathrm{nC}\) is spread uniformly over the surface of one face of a nonconducting disk of radius \(1.25 \mathrm{cm} .\) (a) Find the magnitude and direction of the electric field this disk produces at a point \(P\) on the axis of the disk a distance of 2.00 \(\mathrm{cm}\) from its center. (b) Suppose that the charge were all pushed away from the center and distributed uniformly on the outer rim of the disk. Find the magnitude and direction of the electric field at point \(P .\) (c) If the charge is all brought to the center of the disk, find the magnitude and direction of the electric field at point \(P .\) (d) Why is the field in part (a) stronger than the field in part (b)? Why is the field in part (c) the strongest of the three fields?

Particles in a Gold Ring. You have a pure ( 24 karal) gold ring with mass 17.7 \(\mathrm{g}\) . Gold has an atomic mass of 197 \(\mathrm{g} / \mathrm{mol}\) and an atomic number of \(79 .\) ( a) How many protons are in the ring, and what is their total positive charge? (b) If the ring carries no net charge, how many electrons are in it?

Two small plastic spheres are given positive electrical charges. When they are 15.0 \(\mathrm{cm}\) apart, the repulsive force between them has magnitude 0.220 \(\mathrm{N}\) . What is the charge on each sphere (a) if the two charges are equal and (b) if one sphere has four times the charge of the other?

Two small, copper spheres each have radius 1.00 \(\mathrm{mm}\) . (a) How many atoms does each sphere contain? (b) Assume that each copper atom contains 29 protons and 29 electrons. We know that electrons and protons have charges of exactly the same magnitude, but let's explore the effect of small differences (see also Problem 21.83\()\) . If the charge of a proton is \(+e\) and the magnitude of the charge of an electron is 0.100\(\%\) smaller, what is the net charge of each sphere and what force would one sphere exert on the other if they were separated by 1.00 \(\mathrm{m} ?\)

(a) Sketch the electric field lines for an infinite line of charge. You may find it helpful to show the field lines in a plane containing the line of charge in one sketch and the field lines in a a plane perpendicular to the line of charge in a second sketch. (b) Explain how your sketches show (i) that the magnitude \(E\) of the electric field depends only on the distance \(r\) from the line of charge and (ii) that \(E\) decreases like \(1 / r .\)
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