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Chapter 21: Problem 19

(a) You take an LP record out of its sleeve, and it acquires a static charge of \(1 \mathrm{nC}\). You play it at the normal speed of \(33 \frac{1}{3}\) r.p.m. and the charge moving in a circle creates an electric current. What is the current, in amperes? (b) Although the planetary model of the atom can be made to work with any value for the radius of the electrons' orbits, more advanced models that we will study later in this course predict definite radii. If the electron is imagined as circling around the proton at a speed of \(2.2 \times 10^{6} \mathrm{~m} / \mathrm{s}\), in an orbit with a radius of \(0.05 \mathrm{~nm}\), what electric current is created? The charge of an electron is \(-e=-1.60 \times\) \(10^{-19} \mathrm{C}\)

Short Answer

Expert verified
(a) 0.556 nA, (b) 1.12 mA.

Step by step solution

01

Understand the Problem - LP Record

We need to find the electric current produced by a static charge moving in a circle. The given charge is \(1\, \mathrm{nC}\) (which is \(1 \times 10^{-9}\, \mathrm{C}\)), and the LP record rotates at \(33\frac{1}{3}\) revolutions per minute.
02

Convert RPM to RPS - LP Record

Convert the rotation speed from revolutions per minute to revolutions per second. \[33\frac{1}{3}\, \text{rpm} = \frac{100}{3} \text{ rpm} = \frac{100}{3 \times 60} \text{ rps} = \frac{5}{9} \text{ rps}\]
03

Calculate Electric Current - LP Record

The formula for current \(I\) is \(I = \frac{Q}{T}\), where \(Q\) is the charge and \(T\) is the time period for one revolution. Since the period \(T\) is the inverse of the frequency \(f\), we have \(T = \frac{1}{f}\). Thus, \[I = Q \times f = 1 \times 10^{-9} \times \frac{5}{9}\, \text{A} = \frac{5 \times 10^{-9}}{9}\, \text{A} = 0.556 \times 10^{-9}\, \text{A} = 0.556 \text{ nA}\]
04

Understand the Problem - Electron Orbit

We need to calculate the electric current produced by an electron moving in a circular orbit. The speed is given as \(v = 2.2 \times 10^{6}\, \mathrm{m/s}\), the radius of the orbit is \(0.05 \mathrm{~nm}\), and the electron charge is \( -1.60 \times 10^{-19}\, \mathrm{C}\).
05

Calculate Time Period - Electron Orbit

Find the time period \(T\) for one complete revolution of the electron. The circumference \(C\) of the orbit is \(2\pi r\) where \(r = 0.05 \times 10^{-9}\, \text{m}\) thus, \[C = 2\pi \times 0.05 \times 10^{-9}\, \text{m} = 0.314 \times 10^{-9}\, \text{m}\]. The time period \(T\) is \(\frac{C}{v}\), therefore \[T = \frac{0.314 \times 10^{-9}}{2.2 \times 10^{6}}\, \text{s} = 1.427 \times 10^{-16}\, \text{s}\].
06

Calculate Electric Current - Electron Orbit

Using the formula \(I = \frac{Q}{T}\) for the current, where \(Q = 1.60 \times 10^{-19}\, \text{C}\) is the charge of the electron, we find: \[I = \frac{1.60 \times 10^{-19}}{1.427 \times 10^{-16}} = 1.12 \times 10^{-3}\, \text{A} = 1.12 \text{ mA}\]
07

Summary of Results

For part (a), the current produced by the LP record is approximately \(0.556\, \text{nA}\). For part (b), the current due to the electron orbiting around the proton is approximately \(1.12\, \text{mA}\).

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Key Concepts

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

Static Electricity
Static electricity refers to an imbalance between positive and negative charges in an object. This imbalance causes charges to move, trying to regain equilibrium. Static electricity is prevalent in various daily scenarios, such as rubbing a balloon on your hair or taking an LP record out of its sleeve.
When the LP record in our exercise is removed from its sleeve, it acquires a charge of 1 nanocoulomb (nC). The resulting static charge doesn't flow through the material; it remains on the object's surface, temporarily "stuck" in that position.
But when the record spins, this static charge causes movement, leading to an electric current in a circular path. The process involves transforming static electricity into kinetic energy that manifests as an electric current.
Electron Orbit
Electrons orbit atoms in specific, predictable paths, akin to how planets orbit the sun. While classic models described these as circular paths without precise radii, advanced atomic theories provide fixed values.
In our example of an electron orbiting a proton, it follows a circular path with a radius of 0.05 nm. This setup is critical in deriving many of the electron's properties, such as energy levels and behavior in magnetic fields.
  • Electron speed is given at 2.2 million meters per second.
  • The charge of an electron, which influences the generated electric current, is essential for many calculations.
Understanding these orbits helps scientists predict reactions and interactions at the atomic level.
Charge Movement
Charge movement pertains to the flow or displacement of electric charge. This movement is integral to currents, as it generates the phenomena we utilize in everyday electronics.
In the case of our LP record, once the static charge begins to move (due to the record's rotation at nearly 33.33 revolutions per minute), it produces an electric current. This current occurs due to the sequential displacement of charges during the spinning motion.
  • Charges moving in circular paths can lead to continuous currents.
  • The frequency of revolution directly affects the amount of current generated.
This is similar to how electrons create current while orbiting atomic nuclei. As electrons move in circular paths around protons, they too generate a current based on charge and velocity.
Circular Motion
Circular motion plays a vital role in understanding electric currents and static charges. An object traveling in a circular path experiences consistent angular speed and acceleration, contributing to steady current production.
For LP records, the spinning motion allows for repeated, uniform charge movement, releasing energy in a cyclic manner. The number of revolutions dictates how frequently the charge "completes" a circuit, practically determining current strength.
When thinking about electrons, their circular motion around a proton forms a constant cycle influenced by speed and radius. This movement is crucial to understanding how electromagnetic waves can be emitted from atoms.
  • The formula for current shows it equals charge times frequency, crucially influenced by circular motion dynamics.
  • Circular orbits of electrons serve as a foundation for much of quantum mechanics.
Circular paths are fundamental in connecting mechanical movement to electrical phenomena.

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Most popular questions from this chapter

Wire is sold in a series of standard diameters, called "gauges." The difference in diameter between one gauge and the next in the series is about \(20 \%\). How would the resistance of a given length of wire compare with the resistance of the same length of wire in the next gauge in the series?

You have a circuit consisting of two unknown resistors in series, and a second circuit consisting of two unknown resistors in parallel. (a) What, if anything, would you learn about the resistors in the series circuit by finding that the currents through them were equal? (b) What if you found out the voltage differences across the resistors in the series circuit were equal? (c) What would you learn about the resistors in the parallel circuit from knowing that the currents were equal? (d) What if the voltages in the parallel circuit were equal?

The subatomic particles called muons behave exactly like electrons, except that a muon's mass is greater by a factor of \(206.77\). Muons are continually bombarding the Earth as part of the stream of particles from space known as cosmic rays. When a muon strikes an atom, it can displace one of its electrons. If the atom happens to be a hydrogen atom, then the muon takes up an orbit that is on the average \(206.77\) times closer to the proton than the orbit of the ejected electron. How many times greater is the electric force experienced by the muon than that previously felt by the electron?

Each bulb has a resistance of one ohm. How much power is drawn from the one- volt battery?

(a) Many battery-operated devices take more than one battery. If you look closely in the battery compartment, you will see that the batteries are wired in series. Consider a flashlight circuit. What does the loop rule tell you about the effect of putting several batteries in series in this way? (b) The cells of an electric eel's nervous system are not that different from ours - each cell can develop a voltage difference across it of somewhere on the order of one volt. How, then, do you think an electric eel can create voltages of thousands of volts between different parts of its body?
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