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Chapter 29: Problem 50

Consider a clinical MRI (magnetic resonance imaging) superconducting mag. net has a diameter of \(1.00 \mathrm{~m}\) length of \(1.50 \mathrm{~m}\), and a uniform magnetic field of 3.00 T. Determine (a) the energy density of the magnetic field and (b) the total energy in the solenoid.

Short Answer

Expert verified
Based on the given solution: (a) The energy density of the magnetic field is \(\frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3}\). (b) The total energy in the solenoid is \(\frac{27}{32 \times 10^{-7}} \mathrm{J}\).

Step by step solution

01

Calculate the energy density of the magnetic field

The energy density of a magnetic field is given by the formula \(u = \frac{B^2}{2 \mu_0}\), where \(B\) is the magnetic field and \(\mu_0\) is the permeability of free space. The permeability of free space is a constant, \(\mu_0 = 4 \pi × 10^{-7} \mathrm{T \cdot m/A}\). Using the given magnetic field value of \(B = 3.00 \mathrm{T}\), we can find the energy density: \(u = \frac{(3.00\mathrm{T})^2}{2 \times (4 \pi × 10^{-7} \mathrm{T \cdot m/A})} = \frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3}\)
02

Calculate the volume of the solenoid

The solenoid has a cylindrical shape with a diameter of \(1.00 \mathrm{m}\) and a length of \(1.50 \mathrm{m}\). To calculate its volume, we first need to find the radius (which is half of the diameter) and then use the formula for the volume of a cylinder, \(V = \pi r^2 h\), where \(r\) is the radius and \(h\) is the height (length). The radius is \(0.50 \mathrm{m}\), so the volume of the solenoid is: \(V = \pi (0.50 \mathrm{m})^2 (1.50 \mathrm{m}) = \frac{3}{4}\pi \mathrm{m^3}\)
03

Calculate the total energy in the solenoid

Now that we have the energy density and the volume of the solenoid, we can calculate the total energy stored in the solenoid by simply multiplying the energy density by the volume. \(E = u \times V = \frac{9}{8 \pi × 10^{-7}} \mathrm{J/m^3} \times \frac{3}{4}\pi \mathrm{m^3} = \frac{27}{32 \times 10^{-7}} \mathrm{J}\) Therefore, the total energy in the solenoid is \(\frac{27}{32 \times 10^{-7}} \mathrm{J}\).

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Most popular questions from this chapter

Chapter 14 discussed damped harmonic oscillators, in which the damping force is velocity dependent and always opposes the motion of the oscillator. One way of producing this type of force is to use a piece of metal, such as aluminum, that moves through a nonuniform magnetic field. Explain why this technique is capable of producing a damping force.

A circular coil of wire with 20 turns and a radius of \(40.0 \mathrm{~cm}\) is laying flat on a horizontal table as shown in the figure. There is a uniform magnetic field extending over the entire table with a magnitude of \(5.00 \mathrm{~T}\) and directed to the north and downward, making an angle of \(25.8^{\circ}\) with the horizontal. What is the magnitude of the magnetic flux through the coil?

Your friend decides to produce electrical power by turning a coil of \(1.00 \cdot 10^{5}\) circular loops of wire around an axis parallel to a diameter in the Earth's magnetic field, which has a local magnitude of \(0.300 \mathrm{G}\). The loops have a radius of \(25.0 \mathrm{~cm} .\) a) If your friend turns the coil at a frequency of \(150.0 \mathrm{~Hz}\) what peak current will flow in a resistor, \(R=1500 . \Omega\) connected to the coil? b) The average current flowing in the coil will be 0.7071 times the peak current. What will be the average power obtained from this device?

What is the inductance in a series \(\mathrm{RL}\) circuit in which \(R=3.00 \mathrm{k} \Omega\) if the current increases to one half of its final value in \(20.0 \mu \mathrm{s} ?\)

An emf of \(20.0 \mathrm{~V}\) is applied to a coil with an inductance of \(40.0 \mathrm{mH}\) and a resistance of \(0.500 \Omega\). a) Determine the energy stored in the magnetic field when the current reaches one fourth of its maximum value. b) How long does it take for the current to reach this value?
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