Concept explainers
Review. The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. P23.73) could carry a power of 1.00 × 103 MW (the output of a large power plant) at 200 kV, DC, over a distance of 1.00 × 103 km without loss. An inner wire of radius a = 2.00 cm, made from the superconductor Nb3Sn, carries the current I in one direction. A surrounding superconducting cylinder of radius b = 5.00 cm would carry the return current I. In such a system, what is the magnetic field (a) at the surface of the inner conductor and (b) at the inner surface of the outer conductor? (c) How much energy would be stored in the magnetic field in the space between the conductors in a 1.00 × 103 km superconducting line? (d) What is the pressure exerted on the outer conductor due to the current in the inner conductor?
Figure. P23.73
(a)
The magnetic field at the inner surface of the conductor
Answer to Problem 73P
The magnetic field at the inner surface of the conductor has a value of
Explanation of Solution
Write the equation for the ampere’s law.
Here,
Write the equation for the power on a single cable.
Here,
Conclusion:
Substitute
Therefore, the magnetic field at the inner surface of the conductor has a value of
(b)
The magnetic field at the inner surface of the outer conductor
Answer to Problem 73P
The magnetic field at the inner surface of the outer conductor has a value of
Explanation of Solution
Write the equation for the ampere’s law from equation (I).
Here,
Write the equation for the power on a single cable. From equation (II).
Here,
Conclusion:
Substitute
Therefore, the magnetic field at the inner surface of the outer conductor has a value of
(c)
The energy stored in the magnetic field
Answer to Problem 73P
The energy stored in the magnetic field in the space between the conductors is
Explanation of Solution
Write the equation for the energy density in the magnetic field.
Here,
Write the equation for the energy stored in the magnetic field.
Here,
Write the equation for the change in volume of the conductor.
Here,
Substitute equation (I) and equation (III) in equation (II).
Substitute equation (I) in equation (VII).
Conclusion:
Substitute
Therefore, the energy stored in the magnetic field in the space between the conductors is
(d)
The pressure exerted on the outer conductor
Answer to Problem 73P
The pressure exerted on the outer conductor is
Explanation of Solution
Consider a small rectangular segment with length
Write the equation for the outward force experienced by the rectangular segment.
Here,
Substitute
Write the equation for the pressure exerted on the outer conductor.
Here,
Conclusion:
Substitute equation (X) in equation (XI).
Therefore, the pressure exerted on the outer conductor is
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Chapter 23 Solutions
Principles of Physics: A Calculus-Based Text
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