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Chapter 11: Problem 4

(a) At time \(t=0\), a positively charged particle is placed, at rest, in a vacuum, in which there is a uniform electric field of magnitude \(E\). Write an equation giving the particle's speed, \(v\), in terms of \(t, E\), and its mass and charge \(m\) and \(q\). (answer check available at lightandmatter.com) (b) If this is done with two different objects and they are observed to have the same motion, what can you conclude about their masses and charges? (For instance, when radioactivity was discovered, it was found that one form of it had the same motion as an electron in this type of experiment.)

Short Answer

Expert verified
(a) \( v = \frac{qE}{m} t \). (b) The two objects have the same mass-to-charge ratio: \( \frac{q_1}{m_1} = \frac{q_2}{m_2} \).

Step by step solution

01

Understand the Forces Acting on the Particle

The particle being positively charged will experience a force due to the electric field. The force can be described by the equation \( F = qE \), where \( F \) is the force, \( q \) is the charge of the particle, and \( E \) is the electric field strength.
02

Apply Newton's Second Law

According to Newton's second law, the force acting on an object is equal to the mass of the object times its acceleration (\( F = ma \)). Since the only force acting on the particle is the electric force, we can equate it to \( ma \). So, \( ma = qE \), which gives \( a = \frac{qE}{m} \).
03

Relate Acceleration to Velocity

The particle starts from rest, so initially the velocity \( v = 0 \). To find the velocity at time \( t \), integrate the acceleration with respect to time. The acceleration is constant \( a = \frac{qE}{m} \), so \( v = \int_0^t a \, dt = \int_0^t \frac{qE}{m} \, dt = \frac{qE}{m} t \). Hence, the speed \( v = \frac{qE}{m} t \).
04

Conclusion for Same Motion

When two different objects have the same motion, specifically the same acceleration under the same conditions (same \( E \) and \( t \)), their mass-to-charge ratio must be the same. Therefore, \( \frac{q_1}{m_1} = \frac{q_2}{m_2} \), where \( q_1, m_1 \) are the charge and mass of the first object, and \( q_2, m_2 \) are those of the second object.

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Key Concepts

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

Charge-to-Mass Ratio
The charge-to-mass ratio of a particle is a crucial concept in understanding how charged particles behave in electric fields. It is given by the formula \( \frac{q}{m} \), where \( q \) is the charge of the particle and \( m \) is its mass.

This ratio highlights how a charged particle will respond to an electric field. In a uniform electric field, particles with a higher charge-to-mass ratio will experience greater acceleration compared to those with a lower charge-to-mass ratio.

  • High charge-to-mass ratio means more acceleration.
  • Low charge-to-mass ratio means less acceleration.
When comparing different particles, if they exhibit the same motion under identical conditions, it implies that their charge-to-mass ratios are identical. This fascinating property helped in historical discoveries, such as identifying similar behaviors between electrons and other particles.
Newton鈥檚 Second Law
Newton's Second Law is fundamental in understanding how forces affect an object's motion. It is expressed by the equation \( F = ma \), where \( F \) is the force acting on an object, \( m \) is its mass, and \( a \) is the acceleration caused by the force.

In the context of a charged particle in an electric field, the only force acting on the particle is the electric force. This force can be calculated as \( F = qE \), where \( q \) is the charge of the particle and \( E \) is the magnitude of the electric field.

According to Newton's Second Law, we equate this force to \( ma \), leading us to the equation \( ma = qE \). Simplifying this gives us the formula for acceleration: \( a = \frac{qE}{m} \). This demonstrates how both the charge and the mass of the particle are critical in determining how it accelerates in the field.
Uniform Electric Field
A uniform electric field is a region where the electric field strength \( E \) is constant at all points.

In such a field, the forces experienced by charged particles are uniform. This results in a constant acceleration for any charged particle, assuming no other forces act, such as gravity. The strength of the electric field affects both the force and acceleration, as described by \( F = qE \).

One key characteristic of uniform electric fields is that they produce straight-line motion for charged particles. This predictable behavior is vital for understanding the dynamics of charged particles, whether they are electrons, ions, or subatomic particles.

Understanding the behavior of particles in a uniform electric field aids in many applications, from designing electronic devices to analyzing particle behavior in physics experiments.
Particle Motion in Electric Fields
The motion of particles in electric fields can be predicted and described using the principles of physics.

For a positively charged particle starting from rest in a uniform electric field, it will experience an increase in speed due to constant acceleration. The equation \( v = \frac{qE}{m} t \) gives us the speed \( v \) at time \( t \). This equation signifies that the speed is directly proportional to both the charge-to-mass ratio \( \frac{q}{m} \) and the time spent in the field.

In cases where two distinct particles show identical motion, they must have the same charge-to-mass ratio, reflecting identical acceleration under the same conditions (like the field strength \( E \) and time \( t \)).

This concept bridges the understanding between observable phenomena and their mathematical representation, providing insight into both experimental data and theoretical predictions.

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Most popular questions from this chapter

(a) Use Gauss' law to find the fields inside and outside an infinite cylindrical surface with radius \(b\) and uniform surface charge density \(\sigma\).(answer check available at lightandmatter.com) (b) Show that there is a discontinuity in the electric field equal to \(4 \pi k \sigma\) between one side of the surface and the other, as there should be (see page 628). (c) Reexpress your result in terms of the charge per unit length, and compare with the field of a line of charge. (d) A coaxial cable has two conductors: a central conductor of radius \(a\), and an outer conductor of radius \(b\). These two conductors are separated by an insulator. Although such a cable is normally used for time-varying signals, assume throughout this problem that there is simply a DC voltage between the two conductors. The outer conductor is thin, as in part c. The inner 46. (a) Use Gauss' law to find the fields inside and outside an infinite cylindrical surface with radius \(b\) and uniform surface charge density \(\sigma\).(answer check available at lightandmatter.com) (b) Show that there is a discontinuity in the electric field equal to \(4 \pi k \sigma\) between one side of the surface and the other, as there should be (see page 628). (c) Reexpress your result in terms of the charge per unit length, and compare with the field of a line of charge. (d) A coaxial cable has two conductors: a central conductor of radius \(a\), and an outer conductor of radius \(b\). These two conductors are separated by an insulator. Although such a cable is normally used for time-varying signals, assume throughout this problem that there is simply a DC voltage between the two conductors. The outer conductor is thin, as in part c. The inner

Find the capacitance of the surface of the earth, assuming there is an outer spherical "plate鈥� at infinity. (In reality, this outer plate would just represent some distant part of the universe to which we carried away some of the earth's charge in order to charge up the earth.)(answer check available at lightandmatter.com)

(a) In a series LC circuit driven by a DC voltage \((\omega=0)\), compare the energy stored in the inductor to the energy stored in the capacitor. (b) Carry out the same comparison for an LC circuit that is oscillating freely (without any driving voltage). (c) Now consider the general case of a series LC circuit driven by an oscillating voltage at an arbitrary frequency. Let \(\overline{U_{L}}\) and be the average energy stored in the inductor, and similarly for \(\overline{U_{C}}\). Define a quantity \(u=\overline{U_{C}} /\left(\overline{U_{L}}+\overline{U_{C}}\right)\), which can be interpreted as the capacitor's average share of the energy, while \(1-u\) is the inductor's average share. Find \(u\) in terms of \(L, C\), and \(\omega\), and sketch a graph of \(u\) and \(1-u\) versus \(\omega\). What happens at resonance? Make sure your result is consistent with your answer to part a.(answer check available at lightandmatter.com)

Find every complex number \(z\) such that \(z^{3}=1\). (solution in the pdf version of the book)

(a) Show that the energy in the electric field of a point charge is infinite! Does the integral diverge at small distances, at large distances, or both? \hwhint \\{hwhint:epointinfty\\} [4] (b) Now calculate the energy in the electric field of a uniformly charged sphere with radius \(b\). Based on the shell theorem, it can be shown that the field for \(r>b\) is the same as for a point charge, while the field for \(r
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