Expression of safety factor in terms of magnetic flux

The safety factor given by Eq. (29) is expressed in terms of the components of the magnetic field. The safety factor can also be expressed in terms of the magnetic flux. Define $d \Psi_p$ as the poloidal magnetic flux enclosed by two neighboring closed magnetic surface, then $d \Psi_p$ can be written

$$d \Psi_p = 2 \pi R d x B_p$$ (30)

where $d x$ is perpendicular to the first magnetic surface (so perpendicular to the $\mathbf{B}_p$) and its length is equal to the length from the point on this magnetic surface to the point on the neighbour magnetic surface. Using Eq. (30), the poloidal magnetic field is written as

$$B_p = \frac{1}{2 \pi R} \frac{d \Psi_p}{d x} .$$ (31)

Substituting Eq. (31) into Eq. (29), we obtain

$$q = \frac{1}{2 \pi} \oint \frac{1}{R} \frac{B_{\phi}}{B_p} d \ell... ...x B_{\phi}}{d \Psi_p} d \ell_p = \oint \frac{d x B_{\phi}}{d \Psi_p} d \ell_p .$$ (32)

We know $d \Psi_p$ is a constant independent of the poloidal location, so $d \Psi_p$ can be taken outside the integration to give

$$q = \frac{1}{d \Psi_p} \oint d x B_{\phi} d \ell_p$$ (33)

On the other hand, the toroidal magnetic flux enclosed by the two magnetic surfaces, $d \Psi_t$, is given by

$$d \Psi_t = \oint (B_{\phi} d x) d \ell_p$$ (34)

Using this, Eq. (33) is written as

$$q = \frac{d \Psi_t}{d \Psi_p}$$ (35)

Equation (35) indicates that the safety factor of a magnetic surface is equal to the differential of the toroidal magnetic flux with respect to the poloidal magnetic flux enclosed by the magnetic surface.