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Chapter 18: Problem 43

A small object has a mass of \(3.0 \times 10^{-3} \mathrm{kg}\) and a charge of \(-34 \mu \mathrm{C} .\) It is placed at a certain spot where there is an electric field. When released, the object experiences an acceleration of \(2.5 \times 10^{3} \mathrm{m} / \mathrm{s}^{2}\) in the direction of the \(+x\) axis. Determine the magnitude and direction of the electric field.

Short Answer

Expert verified
The electric field is \( 2.21 \times 10^{5} \text{ N/C} \) in the -x direction.

Step by step solution

01

Understanding Electric Force

To determine the electric field, we must first calculate the electric force acting on the object. The relationship between force, mass, and acceleration is given by Newton's second law: \( F = m \cdot a \).
02

Calculate the Electric Force

Given the object's mass \( m = 3.0 \times 10^{-3} \) kg and its acceleration \( a = 2.5 \times 10^3 \) m/s虏, we calculate the force as follows: \( F = 3.0 \times 10^{-3} \text{ kg} \times 2.5 \times 10^{3} \text{ m/s}^2 = 7.5 \text{ N} \).
03

Understanding the Relationship Between Electric Field and Force

The electric field \( E \) is related to the electric force \( F \) by the equation \( F = qE \), where \( q \) is the charge of the object.
04

Calculate the Electric Field

The charge of the object is given as \(-34 \) \( \mu \text{C} = -34 \times 10^{-6} \text{ C} \). We can calculate the magnitude of the electric field \( E \) using \( E = \frac{F}{|q|} \). Therefore, \( E = \frac{7.5 \text{ N}}{34 \times 10^{-6} \text{ C}} = 2.21 \times 10^{5} \text{ N/C} \).
05

Determine the Direction of the Electric Field

The object's charge is negative. Since the acceleration is in the positive x-axis direction, the electric field must be in the negative direction to exert a force in the positive direction (because force on a negative charge is opposite to the field direction). Therefore, the electric field is oriented in the -x direction.

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Key Concepts

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

Electric Force
Electric force is a fundamental concept in physics that describes how charged particles interact with each other. It's a manifestation of the electromagnetic force, which is one of the four fundamental forces in the universe. The electric force acting on a charged object is directly related to both the electric field present at the location and the object's electric charge.

To find the force exerted on an object in an electric field, we apply the formula \( F = qE \), where \( F \) is the force, \( q \) is the charge, and \( E \) is the electric field.

Key points to remember about electric force include:
  • It acts over a distance鈥攎eaning the objects don't need to be in physical contact for the force to exist.
  • The force can either be attractive or repulsive, depending on the charges involved. Like charges repel, while opposite charges attract.
Newton's Second Law
Newton's Second Law of Motion is a cornerstone in understanding how forces affect objects. It states that the force acting on an object equals the mass of that object multiplied by its acceleration. Mathematically, it is expressed as \( F = m \cdot a \).

This law allows us to predict how objects will move under various forces. By understanding the mass and acceleration, we can determine the amount of force required or exerted.

Key points about Newton's Second Law:
  • Force and acceleration are directly proportional. Increase in force results in an increase in acceleration if mass is unchanged.
  • Mass and acceleration are inversely proportional. Greater mass results in less acceleration for the same amount of force.
  • Direction of the force vector and acceleration vector are the same.
Electric Charge
Electric charge is a property of matter that causes it to experience a force when placed in an electric field. It's the source of the electric force and is a fundamental property of particles.

Charges exist in two types: positive and negative. This property is responsible for the interaction between objects in electrical contexts. Charge is measured in coulombs (C), with the elementary charge (approximately \(1.6 \times 10^{-19} \) C) being the charge of a single proton or electron.

Aspects to remember about electric charge:
  • Opposite charges attract, similar charges repel.
  • Total charge in an isolated system is conserved.
  • The electric field produced by a charge is proportional to the magnitude of the charge.
Acceleration
Acceleration is the rate of change of velocity an object experiences as a result of forces acting on it. In the context of electric fields, acceleration is caused by the electric force acting on a charged particle.

The relationship between force and acceleration is described by Newton's Second Law. In a situation like the original exercise, when a charged object accelerates due to an electric field, it is because of the electric force acting on it.

Important notes on acceleration:
  • It's a vector quantity, meaning it has both magnitude and direction.
  • Positive acceleration increases velocity, while negative acceleration (deceleration) decreases it.
  • Constant acceleration results in linear velocity change, following the equations of motion.

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Most popular questions from this chapter

You and your team are tasked with designing an "electron gun" that operates in a vacuum chamber, the purpose of which is to direct a beam of electrons toward a tiny metallic plate in order to heat it. The electrons in the beam have a speed of \(3.50 \times 10^{7} \mathrm{m} / \mathrm{s}\) and travel in the positive \(x\) direction through the center of a set of deflecting plates (a parallel-plate capacitor) that sets up a uniform electric field in the region between the plates. The target is located \(22.0 \mathrm{cm}\) along the \(x\) -axis from the trailing edge of plates (i.e., the edge closest to the target), and \(11.5 \mathrm{cm}\) above the horizontal (i.e., in the \(+y\) direction). The length of the plates (in the \(x\) direction) is \(2.50 \mathrm{cm}\) (a) In which direction should the electric field between the plates point in order to deflect the electrons towards the target? (b) To what magnitude should you set the electric field so that the electron beam hits the target? (c) After successfully striking the target using your results from (a) and (b), you realized that the target is not heating up to the required temperature. Since the degree of heating depends on the speed of the electrons, you increase the electron speed to \(5.20 \times 10^{7} \mathrm{m} / \mathrm{s} .\) With the electric field setting from (b), will the electrons still be on target? If not, to what value should you set the electric field?

A uniform electric field exists everywhere in the \(x, y\) plane. This electric field has a magnitude of \(4500 \mathrm{N} / \mathrm{C}\) and is directed in the positive \(x\) direction. A point charge \(-8.0 \times 10^{-9} \mathrm{C}\) is placed at the origin. Determine the magnitude of the net electric field at (a) \(x=-0.15 \mathrm{m}\) (b) \(x=+0.15 \mathrm{m}\) and \((\mathrm{c}) \mathrm{y}=+0.15 \mathrm{m}\)

Two very small spheres are initially neutral and separated by a distance of \(0.50 \mathrm{m} .\) Suppose that \(3.0 \times 10^{13}\) electrons are removed from one sphere and placed on the other. (a) What is the magnitude of the electrostatic force that acts on each sphere? (b) Is the force attractive or repulsive? Why?

Suppose you want to determine the electric field in a certain region of space. You have a small object of known charge and an instrument that measures the magnitude and direction of the force exerted on the object by the electric field. (a) The object has a charge of \(+20.0 \mu \mathrm{C}\) and the instrument indicates that the electric force exerted on it is \(40.0 \mu \mathrm{N}\), due east. What are the magnitude and direction of the electric field? (b) What are the magnitude and direction of the electric field if the object has a charge of \(-10.0 \mu \mathrm{C}\) and the instrument indicates that the force is \(20.0 \mu \mathrm{N},\) due west?

A tiny ball (mass \(=0.012 \mathrm{kg}\) ) carries a charge of \(-18 \mu \mathrm{C}\). What electric field (magnitude and direction) is needed to cause the ball to float above the ground?
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