Homework Help is Here – Start Your Trial Now!
Science
Advanced Physics
(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 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).

(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 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).

icon
Related questions
Question
A problem from David J. Griffiths - Introduction to Electrodynamics. Please help me solve (a)
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.
Expert Solution
steps

Step by step

Solved in 2 steps with 2 images

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer