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Chapter 23: Problem 16

Two stationary point charges \(+3.00 \mathrm{nC}\) and \(+2.00 \mathrm{nC}\) are separated by a distance of 50.0 \(\mathrm{cm} .\) An electron is released from rest at a point midway between the two charges and moves along the line connecting the two charges. What is the speed of the electron when it is 10.0 \(\mathrm{cm}\) from the \(+3.00\) -n \(\mathrm{charge}\)?

Short Answer

Expert verified
The electron's speed is approximately \(2.42 \times 10^6 \text{ m/s}\) at 10 cm from the +3.00 nC charge.

Step by step solution

01

Identify the Problem

We need to find the speed of an electron released midway between two point charges, before it reaches 10 cm away from the +3.00 nC charge.
02

Determine Initial and Final Positions

The midpoint is 25 cm from each charge. The final position of the electron is 10 cm from the +3.00 nC charge, which means it is 40 cm from the +2.00 nC charge.
03

Calculate the Electric Potential Energy Change

Use the formula for electric potential energy change: \[\Delta U = k \cdot \left(\frac{q_1 q_e}{r_{1i}} + \frac{q_2 q_e}{r_{2i}} - \frac{q_1 q_e}{r_{1f}} - \frac{q_2 q_e}{r_{2f}}\right)\] where: \(k = 8.99 \times 10^9 \ \text{N}\cdot\text{m}^2/\text{C}^2\),\(q_1 = +3.00 \text{ nC}\),\(q_2 = +2.00 \text{ nC}\),\(q_e = -1.60 \times 10^{-19} \text{ C} (electron's charge)\),\(r_{1i} = 0.25 \text{ m}\), \(r_{2i} = 0.25 \text{ m}\),\(r_{1f} = 0.10 \text{ m}\), \(r_{2f} = 0.40 \text{ m}\). Calculate the values and find \(\Delta U\).
04

Calculate the Kinetic Energy Change

Since the electron is released from rest, its initial kinetic energy is 0. The change in kinetic energy \(\Delta K\) will be equal to the negative change in potential energy, \(\Delta K = -\Delta U\).
05

Find the Final Speed

Using the equation for kinetic energy \(\Delta K = \frac{1}{2}mv^2\), solve for the speed \(v\) after substituting the change in kinetic energy from the potential energy change, and \(m = 9.11 \times 10^{-31} \text{ kg}\) (mass of an electron).

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Key Concepts

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

Point Charges
In the realm of physics, point charges are theoretical charges that are considered to be concentrated into a single point in space. Despite being idealizations, point charges help us simplify and solve complex electrostatic problems.

The exercise deals with two such point charges, one with a charge of +3.00 nC and the other with +2.00 nC. These point charges create an electric field that influences the motion of an electron placed between them. When an electron is at rest between point charges, its movement is dictated by the electric forces exerted by these charges.
  • Point charges exert a force on each other described by Coulomb's Law.
  • The strength of this force depends on the magnitude of the charges and the distance between them.
  • Coulomb's Law is expressed as:
    \[ F = k \frac{|q_1 q_2|}{r^2} \]
    where \( F \) is the force between the charges, \( k \) is Coulomb's constant, and \( r \) is the distance between the charges.
Through such point charges, we can also define an electric potential energy, which will be useful in understanding the kinetic energy changes for the electron in motion.
Kinetic Energy
Kinetic energy (KE) is the energy associated with the motion of an object. For an electron, which is a tiny particle, the kinetic energy tells us how fast it's moving. When we solve problems involving charged particles, understanding kinetic energy is crucial.

Initially, the electron in our exercise is at rest, which means its kinetic energy is zero. As the electron is released, it begins to move under the influence of the electric field created by the point charges. This electric field does work on the electron, converting electric potential energy into kinetic energy.
  • The basic formula for kinetic energy is:
    \[ KE = \frac{1}{2}mv^2 \]
    where \( m \) is the mass of the electron and \( v \) is its velocity.
  • The change in kinetic energy, \( \Delta KE \), is significant as it helps us find the electron's speed at a given point.
  • According to energy conservation, the change in kinetic energy is equal to the negative of the change in electric potential energy:
    \[ \Delta KE = - \Delta U \]
By understanding kinetic energy, we link the movement of the electron to the energy shifts occurring due to its position relative to the point charges.
Electron Motion
Electron motion in an electric field is a fascinating aspect of electromagnetism, describing how electrons travel among electric forces. In our context, an electron starts from rest midway between two point charges and is observed as it moves closer to one charge.

The motion of the electron is influenced by the electric field created by the point charges. The electron accelerates as it experiences a force due to the potential difference between the charges.
  • The initial position is equidistant from the point charges, meaning the net force initially is zero, but as the electron starts to move, it gets influenced non-uniformly.
  • The electron moves towards the +3.00 nC charge, losing electric potential energy while gaining kinetic energy.
  • As the electron approaches the +3.00 nC charge to a position 10 cm from it, this change in position represents a change in its electric potential energy, which subsequently changes its speed.
Understanding how the electron's motion is derived from the attraction and repulsion in the electric field provides clarity on how charged particles behave under electrostatic forces.

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Most popular questions from this chapter

A thin insulating rod is bent into a semicircular arc of radius \(a,\) and a total electric charge \(Q\) is distributed uniformly along the rod. Calculate the potential at the center of curvature of the arc if the potential is assumed to be zero at infinity.

A vacuum tube diode consists of concentric cylindrical electrodes, the negative cathode and the positive anode. Because of the accumulation of charge near the cathode, the electric potential between the electrodes is not a linear function of the position, even with planar geometry, but is given by $$V(x)=C x^{4 / 3}$$ where \(x\) is the distance from the cathode and \(C\) is a constant, characteristic of a particular diode and operating conditions. Assume that the distance between the cathode and anode is 13.0 \(\mathrm{mm}\) and the potential difference between electrodes is 240 \(\mathrm{V}\) . (a) Determine the value of \(C .\) (b) Obtain a formula for the electric field between the electrodes as a function of \(x\) . (c) Determine the force on an electron when the electron is halfway between the electrodes.

Axons. Neurons are the basic units of the nervous system. They contain long tubular structures called axons that propagate electrical signals away from the ends of the neurons. The axon contains a solution of potassium \(\left(\mathrm{K}^{+}\right)\) ions and large negative organic ions. The axon membrane prevents the large ions from leaking out, but the smaller \(\mathrm{K}^{+}\) ions are able to penetrate the membrane to some degree (Fig. E23.37). This leaves an excess negative charge on the inner surface of the axon membrane and an excess positive charge on the outer surface, resulting in a potential difference across the membrane that prevents further \(K^{+}\) ions from leaking out. Measurements show that this potential difference is typically about 70 \(\mathrm{mV}\) . The thickness of the axon membrane itself varies from about 5 to \(10 \mathrm{nm},\) so we'll use an average of 7.5 \(\mathrm{nm}\) . We can model the membrane as a large sheet having equal and opposite charge densities on its faces. (a) Find the electric field inside the axon membrane, assuming (not too realistically) that it is filled with air. Which way does it point: into or out of the axon? (b) Which is at a higher potential: the inside surface or the outside surface of the axon membrane?

Three point charges, which initially are infinitely far apart, are placed at the corners of an equilateral triangle with sides \(d .\) Two of the point charges are identical and have charge \(q .\) If zero net work is required to place the three charges at the corners of the triangle, what must the value of the third charge be?

Electric charge is distributed uniformly along a thin rod of length \(a,\) with total charge \(Q .\) Take the potential to be zero at infinity. Find the potential at the following points (Fig. P23.79): (a) point \(P,\) a distance \(x\) to the right of the rod, and (b) point \(R,\) a distance \(y\) above the right- hand end of the rod. (c) In parts (a) and \((\mathrm{b}),\) what does your result reduce to as \(x\) or \(y\) becomes much larger than \(a\) ?
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