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Problem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if a = oo and J is finite, then ав at = V x (v x B). a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982).

Problem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if a = oo and J is finite, then ав at = V x (v x B). a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982).

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Problem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if o ∞o and J is finite, then 7.3 Maxwell's Equations a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982). R ƏB at S do= =V x (v x B). FIGURE 7.58 (b) Let S be the surface bounded by the loop (P) at time t, and S' a surface bounded by the loop in its new position (P) at time t + dt (see Fig. 7.58). The change in flux is S' dt B(t + dt) da + do = dt S B(t + dt) da - Use VB0 to show that S √₂³ (where R is the "ribbon" joining P and P'), and hence that ав l vdt at dt B(t + dt) da = S B(t). da. da - R S B(t + dt) da B(t + dt) da (for infinitesimal dt). Use the method of Sect. 7.1.3 to rewrite the second inte- gral as $ and invoke Stokes' theorem to conclude that do ав = √₂₁ (³B - VX ( at Together with the result in (a), this proves the theorem. (B x V).dl, 353 - V x (v x B) da.
Transcribed Image Text:Problem 7.63 Prove Alfven's theorem: In a perfectly conducting fluid (say, a gas of free electrons), the magnetic flux through any closed loop moving with the fluid is constant in time. (The magnetic field lines are, as it were, "frozen" into the fluid.) (a) Use Ohm's law, in the form of Eq. 7.2, together with Faraday's law, to prove that if o ∞o and J is finite, then 7.3 Maxwell's Equations a slightly different approach to the same problem, see W. K. Terry, Am. J. Phys. 50, 742 (1982). R ƏB at S do= =V x (v x B). FIGURE 7.58 (b) Let S be the surface bounded by the loop (P) at time t, and S' a surface bounded by the loop in its new position (P) at time t + dt (see Fig. 7.58). The change in flux is S' dt B(t + dt) da + do = dt S B(t + dt) da - Use VB0 to show that S √₂³ (where R is the "ribbon" joining P and P'), and hence that ав l vdt at dt B(t + dt) da = S B(t). da. da - R S B(t + dt) da B(t + dt) da (for infinitesimal dt). Use the method of Sect. 7.1.3 to rewrite the second inte- gral as $ and invoke Stokes' theorem to conclude that do ав = √₂₁ (³B - VX ( at Together with the result in (a), this proves the theorem. (B x V).dl, 353 - V x (v x B) da.
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