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Chapter 30: Problem 15

(I) The magnetic field inside an air-filled solenoid \(38.0 \mathrm{~cm}\) long and \(2.10 \mathrm{~cm}\) in diameter is \(0.600 \mathrm{~T}\). Approximately how much energy is stored in this field?

Short Answer

Expert verified
The energy stored in the magnetic field is approximately 18.9 J.

Step by step solution

01

Understanding the Problem

We need to calculate the energy stored in a magnetic field inside a solenoid. We are given the magnetic field strength \(B = 0.600 \mathrm{~T}\), the length of the solenoid \(L = 38.0 \mathrm{~cm} = 0.38 \mathrm{~m}\), and the diameter \(d = 2.10 \mathrm{~cm} = 0.021 \mathrm{~m}\).
02

Calculating the Volume of the Solenoid

The volume \(V\) of the solenoid is calculated using the formula for the volume of a cylinder: \[ V = \pi r^2 L \] where \(r\) is the radius. Given the diameter \(d = 0.021 \mathrm{~m}\), we find the radius \(r = \frac{d}{2} = \frac{0.021}{2} \mathrm{~m} = 0.0105 \mathrm{~m}\). Thus, \[ V = \pi (0.0105)^2 (0.38) \approx 1.32 \times 10^{-4} \mathrm{~m^3} \]
03

Using the Energy Stored in a Magnetic Field Formula

The energy \(U\) stored in a magnetic field can be given by the formula: \[ U = \frac{1}{2\mu_0} B^2 V \]where \(\mu_0 = 4\pi \times 10^{-7} \mathrm{~T \cdot m/A}\) is the permeability of free space. Substitute \(B = 0.600 \mathrm{~T}\) and \(V \approx 1.32 \times 10^{-4} \mathrm{~m^3}\):\[ U = \frac{1}{2 (4\pi \times 10^{-7})} (0.600)^2 (1.32 \times 10^{-4}) \]
04

Calculating the Energy

Compute the above expression to find the energy stored:\[ U \approx \frac{1}{2 \times 4\pi \times 10^{-7}} (0.36) (1.32 \times 10^{-4}) \approx \frac{1}{8\pi \times 10^{-7}} (0.00004752) \]\[ U \approx \frac{0.00004752}{8 \times 3.14159 \times 10^{-7}} \approx \frac{0.00004752}{2.513274122 \times 10^{-6}} \approx 18.9 \mathrm{~J} \]
05

Conclusion

After performing the calculations, the energy stored in the magnetic field inside the solenoid is approximately \(18.9 \mathrm{~J}\).

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Key Concepts

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

Solenoid
A solenoid is essentially a coil of wire that is wrapped in a cylindrical shape. When electric current flows through the wire, it generates a magnetic field. The strength of this field is dependent on factors like the number of turns in the coil and the current passing through it. Solenoids play a crucial role in creating controlled magnetic fields, which makes them valuable in numerous applications.

Key aspects of solenoids include:
  • They can convert electrical energy into mechanical work, often acting as a simple electromagnet.
  • They are commonly used in various devices, like electromechanical actuators and inductors.
  • The magnetic field within a solenoid is strongest at its core and weakens as you move toward the ends.
In the exercise above, understanding the physical dimensions of the solenoid, like its length and diameter, is important in calculating the volume and thus the magnetic energy stored.
Magnetic Field
A magnetic field is a vector field that surrounds magnets and moving charges. It represents the influence that a magnet or a moving charge exerts in the space around it. The magnetic field is denoted by the symbol \(B\) and is measured in teslas (\(T\)).

For solenoids:
  • The magnetic field inside a solenoid is approximately uniform and can be expressed as \(B = \mu_0 nI\), where \(n\) is the turn density and \(I\) is the current.
  • Outside the solenoid, the magnetic field is much weaker because the field lines are concentrated inside.
  • The density of the field lines indicates the strength of the magnetic field 鈥� more concentrated lines equal a stronger field.
In magnetic energy storage, as demonstrated in the solution, the energy stored relies heavily on the intensity of this magnetic field.
Electromagnetism
Electromagnetism is the force responsible for the behavior of electric and magnetic fields. It describes how electric currents create magnetic fields and vice versa. This is the basis for understanding how solenoids and electromagnetic devices function.

Important points on electromagnetism include:
  • Maxwell's equations are the foundation of electromagnetism, helping us understand how electric and magnetic fields interact.
  • In solenoids, electricity is converted into magnetism, showcasing a primary application of electromagnetic principles.
  • The energy stored in the magnetic field of devices like solenoids can be calculated using specific formulas, highlighting the interconnected nature of electrical and magnetic energies.
This concept allows us to build practical technology, such as inductors and transformers, which are critical in electrical circuits and magnetic energy storage solutions.

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Most popular questions from this chapter

(a) What is the rms current in an \(R C\) circuit if \(R=5.70 \mathrm{k} \Omega\), \(C=1.80 \mu \mathrm{F},\) and the rms applied voltage is \(120 \mathrm{~V}\) at \(60.0 \mathrm{~Hz} ?\) (b) What is the phase angle between voltage and current? (c) What is the power dissipated by the circuit? (d) What are the voltmeter readings across \(R\) and \(C\) ?

A \(10.0-\mathrm{k} \Omega\) resistor is in series with a \(26.0-\mathrm{mH}\) inductor and an ac source. Calculate the impedance of the circuit if the source frequency is (a) \(55.0 \mathrm{~Hz}\); (b) \(55,000 \mathrm{~Hz}\).

In a plasma globe, a hollow glass sphere is filled with low-pressure gas and a small spherical metal electrode is located at its center. Assume an ac voltage source of peak voltage \(V_{0}\) and frequency \(f\) is applied between the metal sphere and the ground, and that a person is touching the outer surface of the globe with a fingertip, whose approximate area is \(1.0 \mathrm{~cm}^{2}\). The equivalent circuit for this situation is shown in Fig. \(30-35,\) where \(R_{\mathrm{G}}\) and \(R_{\mathrm{p}}\) are the resistances of the gas and the person, respectively, and \(C\) is the capacitance formed by the gas, glass, and finger. (a) Determine \(C\) assuming it is a parallel-plate capacitor. The conductive gas and the person's fingertip form the opposing plates of area \(A=1.0 \mathrm{~cm}^{2}\). The plates are separated by glass (dielectric constant \(K=5.0\) ) of thickness \(d=2.0 \mathrm{~mm} .\) (b) In a typical plasma globe, \(f=12 \mathrm{kHz}\) Determine the reactance \(X_{C}\) of \(C\) at this frequency in \(\mathrm{M} \Omega\). (c) The voltage may be \(V_{0}=2500 \mathrm{~V}\). With this high voltage, the dielectric strength of the gas is exceeded and the gas becomes ionized. In this "plasma" state, the gas emits light ("sparks") and is highly conductive so that \(R_{\mathrm{G}} \ll X_{C}\). Assuming also that \(R_{\mathrm{p}} \ll X_{C},\) estimate the peak current that flows in the given circuit. Is this level of current dangerous? \((d)\) If the plasma globe operated at \(f=1.0 \mathrm{MHz},\) estimate the peak current that would flow in the given circuit. Is this level of current dangerous?

(II) A damped \(L C\) circuit loses 3.5\(\%\) of its electromagnetic energy per cycle to thermal energy. If \(L = 65 \mathrm { mH }\) and \(C = 1.00 \mu \mathrm { F } ,\) what is the value of \(R ?\)

(I) What is the reactance of a \(9.2 - \mu \mathrm { F }\) capacitor at a frequency of \(( a ) 60.0 \mathrm { Hz } , ( b ) 1.00 \mathrm { MHz }\) ?
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