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Chapter 23: Problem 36

Accelerated Electron An electron is accelerated eastward at \(1.80 \times 10^{9} \mathrm{~m} / \mathrm{s}^{2}\) by an electric field. Determine the magnitude and direction of the electric field.

Short Answer

Expert verified
The magnitude is \( 10.2 \text{ N/C} \) and the direction is westward.

Step by step solution

01

- Identify the given values

The acceleration of the electron is given as: \( a = 1.80 \times 10^9 \text{ m/s}^2 \)
02

- Recall the relationship between force, electric field, and charge

The electric force \( F \) on a charge \( q \) in an electric field \( E \) is given by the equation: \( F = qE \)
03

- Use Newton's second law of motion

According to Newton's second law: \( F = ma \) Where \( m \) is the mass of the electron and \( a \) is its acceleration.
04

- Equate the two expressions for force

From Step 2 and Step 3, we have: \( qE = ma \)
05

- Solve for the electric field \( E \)

Rearrange the equation to solve for \( E \): \( E = \frac{ma}{q} \)
06

- Substitute the known values into the equation

The charge of the electron is \( q = -1.60 \times 10^{-19} \text{ C} \) and the mass of the electron is \( m = 9.11 \times 10^{-31} \text{ kg} \). Substituting these values along with \( a = 1.80 \times 10^9 \text{ m/s}^2 \) into the equation: \( E = \frac{(9.11 \times 10^{-31} \text{ kg})(1.80 \times 10^9 \text{ m/s}^2)}{-1.60 \times 10^{-19} \text{ C}} \)
07

- Calculate the electric field

Perform the calculation: \( E = \frac{1.6398 \times 10^{-21}}{-1.60 \times 10^{-19}} \) \( E = -1.02 \times 10^1 \text{ N/C} \) Since the electric field is in the direction opposite to the force on a negative charge, it points westward.

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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 crucial concept in electrostatics. It describes the force exerted by an electric field on a charged particle. Mathematically, this force is expressed as: \( F = qE \)
where:
  • \(F\) is the electric force
  • \(q\) is the charge of the particle
  • \(E\) is the electric field strength
A positive charge experiences a force in the direction of the electric field, whereas a negative charge, like an electron, experiences a force opposite to the field鈥檚 direction.
Newton's Second Law
Newton's second law is fundamental to understanding the motion of particles under various forces. It states that: \( F = ma \)
This means that a force acting on a mass \(m\) produces an acceleration \(a\). By linking this to the electric force, we can write:
\( ma = qE \)
This relationship allows us to determine how an electron accelerates when subjected to an electric field. By knowing the electron鈥檚 mass and acceleration, we can solve for the electric field strength by rearranging the equation:
\( E = \frac{ma}{q} \)
Electric Field Direction
The direction of the electric field is a significant consideration, as it affects how charges move. An electric field exerts a force on charges in a specific direction:
  • Positive charges are driven along the direction of the field.
  • Negative charges, such as electrons, move opposite to the field direction.
In our example, since the electron is accelerating eastward, the electric force must act towards the east. However, the electric field itself must point westward because the electron, being negatively charged, moves opposite to the electric field direction.

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Most popular questions from this chapter

High-Speed Protons Beams of high-speed protons can be produced in "guns" using electric fields to accelerate the protons. (a) What acceleration would a proton experience if the gun's electric field were \(2.00 \times 10^{4} \mathrm{~N} / \mathrm{C} ?(\mathrm{~b})\) What speed would the proton attain if the field accelerated the proton through a distance of \(1.00 \mathrm{~cm}\) ?

Velocity Components At some instant the velocity components of an electron moving between two charged parallel plates are \(v_{x}=\) \(1.5 \times 10^{5} \mathrm{~m} / \mathrm{s}\) and \(v_{y}=3.0 \times 10^{3} \mathrm{~m} / \mathrm{s}\). Suppose that the electric field between the plates is given by \(\vec{E}=(120 \mathrm{~N} / \mathrm{C}) \hat{\mathrm{j}} .\) (a) What is the acceleration of the electron? (b) What will be the velocity of the electron after its \(x\) coordinate has changed by \(2.0 \mathrm{~cm}\) ?

Dipole in a Field An electric dipole, consisting of charges of magnitude \(1.50 \mathrm{nC}\) separated by \(6.20 \mu \mathrm{m}\), is in an electric field of strength \(1100 \mathrm{~N} / \mathrm{C}\). (a) What is the magnitude of the electric dipole moment? (b) What is the difference between the potential energies corresponding to dipole orientations parallel to and antiparallel to the field?

Torque on a Dipole An electric dipole consists of charges \(+2 e\) and \(-2 e\) separated by \(0.78 \mathrm{~nm}\). It is in an electric field of strength \(3.4 \times 10^{6} \mathrm{~N} / \mathrm{C}\). Calculate the magnitude of the torque on the dipole when the dipole moment is (a) parallel to, (b) perpendicular to, and (c) antiparallel to the electric field.

Functional Dependence and the Electric Field (a) Suppose you want to purchase a sweater in Maryland that has a list price of \(\$ 40\) for which you pay \(\$ 2\) in sales tax. Your friend bought the same sweater in Maryland, but it had a list price of \(\$ 80\) for which she paid \(\$ 4\) in sales tax. How does the ratio of sales tax to price of the sweater compare for you and your friend [i.e., compare the ratios (sales tax)/(sweater price)]? What does that ratio tell us? As what is that ratio defined? (b) Suppose a charge exerts a repulsive force of \(4 \mathrm{~N}\) on a test charge of \(0.2 \mu \mathrm{C}\) that is \(2 \mathrm{~cm}\) from it. However, the charge exerts a repulsive force of \(8 \mathrm{~N}\) on a test charge of \(0.4 \mu \mathrm{C}\) that is \(2 \mathrm{~cm}\) from it. How does the ratio of the force on the test charge to the test charge itself compare in each case [i.e., compare (force felt by test charge)/(test charge)]? What does that ratio tell us? What is that ratio defined as? (c) Suppose a charge \(Q\) exerts a force \(F\) on a test charge \(q\) that is placed near it. By how much would the force exerted by \(Q\) increase if the test charge increased by a factor of \(\alpha\), where \(\alpha\) can be any constant (i.e., \(\alpha=-17\) or 5 or \(7.812\), etc.)? By how much would the ratio of the force on the test charge to the test charge itself increase if the test charge increased by a factor of \(\alpha\) ?Explain. (d) When the value of one quantity depends on the value of a second quantity (and perhaps on others), we say that the first quantity is \(a\) function of the second. How the first quantity changes when the second changes is called the functional dependence. For example, if \(t=A s\), we say that \(t\) has a linear functional dependence on \(s\). When \(s\) doubles, so does \(t\). If \(s\) is divided by 10 , so is \(t\). As a second example, if we had \(y=B x^{2}\), we would say that \(y\) depends quadratically on \(x\). If \(x\) doubles, \(y\) quadruples. If \(x\) is divided by 10, then \(y\) is divided by 100 . (Try this with some numbers, picking whatever values of the constants \(A\) and \(B\) you would like.) (i) What is the functional dependence on the sales tax paid on the price of the sweater in part (a)? Explain. Write an equation that relates the tax paid \((t)\) to the cost of the sweater \((s)\). (ii) What is the functional dependence of the sales tax percentage rate on the price of the sweater in part (a)? Explain. (iii) In part (c), what is the functional dependence of the force magnitude, \(F\), on the amount of the test charge, \(|q| ?\) Explain. (iv) In part (c), what is the functional dependence of the electric field magnitude established by \(Q, E_{Q}\), on the test charge, \(q ?\) Explain.
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