Q63P - Introduction to Electrodynamics

Question

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 $\mathbf{\sigma }\mathbf{=}\mathbf{\infty }$and is J finite, then

                                                             $\frac{\mathbf{\partial }\mathbf{B}}{\mathbf{\partial }\mathbf{t}}\mathbf{=}\mathbf{\nabla }\mathbf{\times }\mathbf{(}\mathbf{v}\mathbf{\times }\mathbf{B}\mathbf{)}$ 

                                   

(b) Let S be the surface bounded by the loop $\left(P\right)$ at time t , and $\mathbf{S}\mathbf{\text{'}}$ a surface bounded by the loop in its new position $\left(P\text{'}\right)$ at time t+dt (see Fig. 7.58). The change in flux is

                                                       $\mathbf{d\Phi }\mathbf{=}{\mathbf{\int }}_{\mathbf{S}\mathbf{\text{'}}}\mathbf{B}\left(t+\mathrm{dt}\right)\mathbf{‐}\mathbf{da}\mathbf{-}{\mathbf{\int }}_{\mathbf{S}}\mathbf{B}\left(t\right)\mathbf{‐}\mathbf{da}$

 

Use $\mathbf{\nabla }\mathbf{·}\mathbf{B}\mathbf{=}\mathbf{0}$ to show that 

                             ${\mathbf{\int }}_{\mathbf{S}\mathbf{\text{'}}}\mathbf{B}\left(t+\mathrm{dt}\right)\mathbf{‐}\mathbf{da}\mathbf{+}{\mathbf{\int }}_{\mathbf{R}}\mathbf{B}\left(t+\mathrm{dt}\right)\mathbf{‐}\mathbf{da}\mathbf{=}{\mathbf{\int }}_{\mathbf{S}}\mathbf{B}\left(t+\mathrm{dt}\right)\mathbf{‐}\mathbf{da}$

(Where R is the "ribbon" joining P and P' ), and hence that 

                                              $\mathbf{d\Phi }\mathbf{=}\mathbf{dt}{\mathbf{\int }}_{\mathbf{S}}\frac{\mathbf{\partial }\mathbf{B}}{\mathbf{\partial }\mathbf{t}}\mathbf{·}\mathbf{da}\mathbf{-}{\mathbf{\int }}_{\mathbf{R}}\mathbf{B}\left(t+\mathrm{dt}\right)\mathbf{‐}\mathbf{da}$

(For infinitesimal dt ). Use the method of Sect. 7.1.3 to rewrite the second integral as 

                                                              $\mathbf{dt}{\mathbf{\oint }}_{\mathbf{P}}\left(B\times v\right)\mathbf{·}\mathbf{dI}$

And invoke Stokes' theorem to conclude that 

                                           $\frac{\mathbf{d\Phi }}{\mathbf{dt}}\mathbf{=}{\mathbf{\int }}_{\mathbf{S}}\left(\frac{\partial B}{\partial t}-\nabla \times \left(v\times B\right)\right)\mathbf{·}\mathbf{da}$

Together with the result in (a), this proves the theorem.

Step-by-Step Solution

Answer

(a) The value to prove that $\frac{\partial \mathrm{B}}{\partial \mathrm{t}}=\nabla \times (\mathrm{v}\times \mathrm{B})$.

(b) The value of invoke stokes’ theorem is $\frac{d\varphi }{dt}={\int }_S\left[\frac{\partial B}{\partial t}-\nabla \times \left(v\times B\right)\right]·\mathrm{da}=0$ .

Step 1: Write the given data from the question.

Let S be the surface bounded by the loop (P) at time t .

Let S' a surface bounded by the loop in its new position (P') at time t+dt .

Let R is the "ribbon" joining P and P' .

Step 2: Determine the formula of ohm’s law using faraday’s law and formula of invoke stokes’ theorem.

Write the formula of ohm’s law using faraday’s law.

$\mathbf{J}\mathbf{=}\mathbf{\sigma }\left(E+v\times B\right)$                                                                                …… (1)

Here, $\mathit{\sigma }$ is charge density, E is electrical field and B is magnetic field. 

Write the formula of invoke stokes’ theorem.

$\mathbf{\int }\mathbf{B}\mathbf{·}\mathbf{da}\mathbf{=}\mathbf{0}$                                                                                        …… (2)

Here, B is magnetic field and da is the radius of the circle.

Step 3: (a) Determine the value to prove that ∂ B ∂ t = ∇ × ( ∇ × B ) .

According to ohm’s law 

Substitute 0 for J and $\infty$ for $\sigma$ into equation (1).

$\left(\mathrm{E}+\mathrm{v}\times \mathrm{B}\right)=0$

 

Taking curl on both sides then

 $\nabla \times \mathrm{E}+\nabla \times \left(\mathrm{v}\times \mathrm{B}\right)=0$

From faraday’s law 

 

$\nabla \times E=-\frac{\partial B}{\partial t}$

Then:

$$\begin{gathered} -\frac{\partial B}{\partial t}+\nabla \times \left(\nabla \times B\right)=0 \\ \frac{\partial B}{\partial t}=\nabla \times \left(\nabla \times B\right) \end{gathered}$$

Therefore, the value to prove that $\frac{\partial B}{\partial t}=\nabla \times \left(\nabla \times B\right)$.

Step 4: (b) Determine the value of invoke stokes’ theorem.

                                           

As we know that for any closed surface.

$\nabla \times \mathrm{B}=0$ 

Determine invoke stokes’ theorem.

Substitute ${\int }_{S\text{'}}B\left(t+dt\right)·da-{\int }_{R}B\left(t+dt\right)·da-{\int }_{S}B\left(t\right).da \mathrm{for} \int \mathrm{B}$ into equation (2).

${\int }_{S\text{'}}B\left(t+dt\right)·da-{\int }_{R}B\left(t+dt\right)·da-{\int }_{S}B\left(t\right).da=0$ 

Here:

$$\begin{gathered} d\varphi ={\int }_{S\text{'}}B\left(t+dt\right)·da-{\int }_{S}B\left(t\right)·da-{\int }_{R}B\left(t+dt\right)·da \\ ={\int }_S\left[B\left(t+dt\right)·da-B\left(t\right)\right]·da-{\int }_{R}B\left(t+dt\right)·da \\ =\left\{{\int }_S\left(\frac{\partial B}{dt}·dt\right)dt-{\int }_{R}B\left(t+dt\right)·da\right\} \\ ={\int }_S\left(\frac{\partial B}{dt}·dt\right)dt-{\int }_{R}B\left(t+dt\right)·\left(\mathrm{dI}\times \mathrm{v}\right)dt \end{gathered}$$ 

Solve further as

Then:

$$\begin{gathered} d\varphi =dt{\int }_S\frac{\partial B}{\partial t}·da-dt{\int }_{S}B·\left(\mathrm{dI}\times \mathrm{v}\right) \\ d\varphi =dt\left\{{\int }_S\left(\frac{\partial B}{\partial t}\right)·da-{\int }_S\nabla \times \left(\mathrm{v}\times \mathrm{B}\right)·da\right\} \\ \frac{d\varphi }{dt}={\int }_S\left[\frac{\partial B}{\partial t}-\nabla \times \left(v\times B\right)\right]·da=0 \end{gathered}$$