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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

A \(100 .-V\) battery is connected in series with a \(500 .-\Omega\) resistor. According to Faraday's Law of Induction, current can never change instantaneously, so there is always some "stray" inductance. Suppose the stray inductance is \(0.200 \mu \mathrm{H}\). How long will it take the current to build up to within \(0.500 \%\) of its final value of \(0.200 \mathrm{~A}\) after the resistor is con- nected to the battery?

Which of the following will induce a current in a loop of wire in a uniform magnetic field? a) decreasing the strength of the field b) rotating the loop about an axis parallel to the field c) moving the loop within the field d) all of the above e) none of the above

A respiration monitor has a flexible loop of copper wire, which wraps about the chest. As the wearer breathes, the radius of the loop of wire increases and decreases. When a person in the Earth's magnetic field (assume \(\left.0.426 \cdot 10^{-4} \mathrm{~T}\right)\) inhales, what is the average current in the loop, assuming that it has a resistance of \(30.0 \Omega\) and increases in radius from \(20.0 \mathrm{~cm}\) to \(25.0 \mathrm{~cm}\) over \(1.00 \mathrm{~s}\) ? Assume that the magnetic field is perpendicular to the plane of the loop.

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?

Two parallel conducting rails with negligible resistance are connected at one end by a resistor of resistance \(R\), as shown in the figure. The rails are placed in a magnetic field \(\vec{B}_{\text {ext }},\) which is perpendicular to the plane of the rails. This magnetic field is uniform and time independent. The distance between the rails is \(\ell\). A conducting rod slides without friction on top of the two rails at constant velocity \(\vec{v}\). a) Using Faraday's Law of Induction, calculate the magnitude of the potential difference induced in the moving rod. b) Calculate the magnitude of the induced current in the \(\operatorname{rod}, i_{\text {ind }}\). c) Show that for the rod to move at a constant velocity as shown, it must be pulled with an external force, \(\vec{F}_{\mathrm{ext}},\) and calculate the magnitude of this force. d) Calculate the work done, \(W_{\text {ext }},\) and the power generated, \(P_{\text {ext }}\), by the external force in moving the rod. e) Calculate the power used (dissipated) by the resistor, \(P_{\mathrm{R}}\). Explain the correlation between this result and those of part (d).
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