Concept explainers
Review. The use of superconductors has been proposed for power transmission lines. A single coaxial cable (Fig. P31.47) 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 he 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 P31.47
(a)
Answer to Problem 47AP
Explanation of Solution
Given info: The power carry by the coaxial cable is
Formula to calculate the current flow in the coaxial cable is,
Here,
Substitute
Thus, the current flow in the coaxial cable is
Formula to calculate the magnetic field at inner conductor from Ampere’s law is,
Here,
Substitute
Conclusion:
Therefore, the magnetic field at the surface of the inner conductor is
(b)
Answer to Problem 47AP
Explanation of Solution
Given info: The power carry by the coaxial cable is
Formula to calculate the magnetic field at inner surface of outer conductor from Ampere’s law is,
Here,
Substitute
Conclusion:
Therefore, the magnetic field at the surface of the inner conductor is
(c)
Answer to Problem 47AP
Explanation of Solution
Given info: The power carry by the coaxial cable is
Formula to calculate the energy density store in magnetic field is,
Formula to calculate the total energy stored in the magnetic field in the space between the conductors is,
Here,
Write the expression for the small arbitrary volume.
Here,
Formula to calculate the magnetic field from Ampere’s law is,
Substitute
Integrate the above equation within limits.
Substitute
Conclusion:
Therefore, the energy that stored in the magnetic field in the space between the conductors is
(d)
Answer to Problem 47AP
Explanation of Solution
Given info: The power carry by the coaxial cable is
The magnetic field created by the inner conductor exerts a force of repulsion on the current in the outer sheath. The strength of this magnetic field is calculated in part (b) that is
Write the expression for the projection area of the outer conductor.
Write the expression for the circumferential area of the outer conductor.
Formula to calculate the current flow in the outer cylinder is,
Here,
Substitute
Substitute
Formula to calculate the force experience by the outer conductor is,
Formula to calculate the pressure exerted on the conductor due to the current is,
Substitute
Substitute
Conclusion:
Therefore, the pressure exerted on the conductor due to the current in the inner conductor is
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Chapter 31 Solutions
Physics for Scientists and Engineers
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