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## Main Question or Discussion Point

Alfven's theorem:

The change of flux through any closed loop moving through a magnetic field of conducting material is zero.

Imagine a loop S moving with speed v and a time later it is a little ahead. It is now loop S', it's dimensions have somehow changed. The "band" which connects S to S' is R. All this happens in a time dt. Initially, the field was B(t). After some time, it is B(t+dt).

(a) J = [tex]\sigma[/tex](E + v X B); J finite, [tex]\sigma[/tex] --> [tex]\infty[/tex], E + (v XB) = O.Take the curl: [tex]\nabla[/tex]xE + [tex]\nabla[/tex]x(v XB) = O. But

8B 8B

Faraday's law says [tex]\nabla[/tex]xE = -dB/dt. So dB/dt = [tex]\nabla[/tex]x(v XB). qed

(b) [tex]\nabla[/tex].B = 0 --> [tex]\oint[/tex] B. da = a for any closed surface. Apply this at time (t + dt) to the surface consisting of

S, S', and R:

[tex]\int[/tex]

(the sign change in the third term comes from switching outward da to inward da).

d[tex]\phi[/tex]=[tex]\int[/tex]

d[tex]\phi[/tex]= [tex]\int[/tex]dB/dt . da} dt -[tex]\int[/tex] B(t + dt) . [(dl X v)

d[tex]\phi[/tex]=dt [tex]\int[/tex] (dB/dt.da) - [tex]\int[/tex]B.(dlxv) [this term is the same as (vxB).dl] =dt{[tex]\int[/tex]dB/dt .da - [tex]\nabla[/tex]x(vXB).da}.

So,

d[tex]\phi[/tex]/dt = 0

Now, I don't understand the part in bold. What does first order and second order refer to?

The change of flux through any closed loop moving through a magnetic field of conducting material is zero.

Imagine a loop S moving with speed v and a time later it is a little ahead. It is now loop S', it's dimensions have somehow changed. The "band" which connects S to S' is R. All this happens in a time dt. Initially, the field was B(t). After some time, it is B(t+dt).

(a) J = [tex]\sigma[/tex](E + v X B); J finite, [tex]\sigma[/tex] --> [tex]\infty[/tex], E + (v XB) = O.Take the curl: [tex]\nabla[/tex]xE + [tex]\nabla[/tex]x(v XB) = O. But

8B 8B

Faraday's law says [tex]\nabla[/tex]xE = -dB/dt. So dB/dt = [tex]\nabla[/tex]x(v XB). qed

(b) [tex]\nabla[/tex].B = 0 --> [tex]\oint[/tex] B. da = a for any closed surface. Apply this at time (t + dt) to the surface consisting of

S, S', and R:

[tex]\int[/tex]

_{S'}B(t + dt) .da + [tex]\int[/tex]_{R}B(t + dt) .da - [tex]\int[/tex]_{S}B(t + dt) .da =a(the sign change in the third term comes from switching outward da to inward da).

d[tex]\phi[/tex]=[tex]\int[/tex]

_{S'}B(t + dt) .da - [tex]\int[/tex]_{S}B(t) .da = [tex]\int[/tex] [B(t + dt) - B(t)] .da - [tex]\int[/tex]_{R}B(t + at) .da ~d[tex]\phi[/tex]= [tex]\int[/tex]dB/dt . da} dt -[tex]\int[/tex] B(t + dt) . [(dl X v)

**Since the second term is already first order in dt, we can replace B(t + dt) by B(t) (the distinction would be**

second order):second order)

d[tex]\phi[/tex]=dt [tex]\int[/tex] (dB/dt.da) - [tex]\int[/tex]B.(dlxv) [this term is the same as (vxB).dl] =dt{[tex]\int[/tex]dB/dt .da - [tex]\nabla[/tex]x(vXB).da}.

So,

d[tex]\phi[/tex]/dt = 0

Now, I don't understand the part in bold. What does first order and second order refer to?

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