Contours of on plane are magnetic surfaces

Contours of $\Psi$ on plane are magnetic surfaces

Because $\Psi$ is constant along a magnetic field line and $\Psi$ is independent of $\phi$ , it follows that the projection of a magnetic field line onto plane is a contour of $\Psi$ . On the other hand, it is ready to prove that the contour of $\Psi$ on plane is also the projection of a magnetic field line onto the plane, i.e., a magnetic surface. [Proof. The contour of $\Psi$ on plane is written

$\displaystyle d \Psi = 0,$

(13)

i.e.,

$\displaystyle \frac{\partial \Psi}{\partial R} d R + \frac{\partial \Psi}{\partial Z} d Z = 0.$

(14)

$\displaystyle \Rightarrow \frac{1}{R} \frac{\partial \Psi}{\partial R} d R + \frac{1}{R} \frac{\partial \Psi}{\partial Z} d Z = 0.$

(15)

Using Eqs. (4) and (5), the above equation is written

$\displaystyle B_Z d R - B_R d Z = 0,$

(16)

i.e.,

$\displaystyle \frac{d Z}{d R} = \frac{B_Z}{B_R},$

(17)

which is the equation of the projection of a magnetic field line on

plane. Thus, we prove that the contour of $\Psi$ is also the projection of a magnetic field line on

plane.]

yj 2018-03-09