Correct the loop-source prefactor and regenerate its figures

book/3_diffusion_fields/em_diffusion.md

@@ -349,14 +349,14 @@ which is a line integral along the four line segments of the loop. The unit vect
 
 $$
 \begin{aligned}
-E_x(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{t^3}}\left\{\exp\left(-\frac{\sigma\mu[(y+L_y/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(y-L_y/2)^2+z^2]}{4t}\right)\right\} \\
+E_x(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{4\pi t^3}}\left\{\exp\left(-\frac{\sigma\mu[(y+L_y/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(y-L_y/2)^2+z^2]}{4t}\right)\right\} \\
 &\quad\times\int_{-L_x/2}^{L_x/2}\exp\left(-\frac{\sigma\mu(x-x')^2}{4t}\right)\mathrm{d}x',
 \end{aligned}
 $$
 
 $$
 \begin{aligned}
-E_y(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{t^3}}\left\{\exp\left(-\frac{\sigma\mu[(x-L_x/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(x+L_x/2)^2+z^2]}{4t}\right)\right\} \\
+E_y(\boldsymbol r,t) &= -\frac{\mu}{4\pi}\sqrt{\frac{\sigma\mu}{4\pi t^3}}\left\{\exp\left(-\frac{\sigma\mu[(x-L_x/2)^2+z^2]}{4t}\right) - \exp\left(-\frac{\sigma\mu[(x+L_x/2)^2+z^2]}{4t}\right)\right\} \\
 &\quad\times\int_{-L_y/2}^{L_y/2}\exp\left(-\frac{\sigma\mu(y-y')^2}{4t}\right)\mathrm{d}y',
 \end{aligned}
 $$
@@ -389,11 +389,11 @@ $$
 where $\mathrm{erf}$ denotes the error function and the scale factor is chosen such that $\mathrm{erf}(\infty)=1$. Substituting these results in the expressions for the electric field, we finally obtain
 
 $$
-E_x(\boldsymbol r,t) = -\frac{\mu}{8\sqrt{\pi}\,t}\left\{\exp\left(-\frac{y_p^2+z^2}{D^2}\right) - \exp\left(-\frac{y_m^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{x_p}{D}\right) - \mathrm{erf}\left(\frac{x_m}{D}\right)\right],
+E_x(\boldsymbol r,t) = -\frac{\mu}{8\pi t}\left\{\exp\left(-\frac{y_p^2+z^2}{D^2}\right) - \exp\left(-\frac{y_m^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{x_p}{D}\right) - \mathrm{erf}\left(\frac{x_m}{D}\right)\right],
 $$ (eq:Exloop)
 
 $$
-E_y(\boldsymbol r,t) = -\frac{\mu}{8\sqrt{\pi}\,t}\left\{\exp\left(-\frac{x_m^2+z^2}{D^2}\right) - \exp\left(-\frac{x_p^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{y_p}{D}\right) - \mathrm{erf}\left(\frac{y_m}{D}\right)\right].
+E_y(\boldsymbol r,t) = -\frac{\mu}{8\pi t}\left\{\exp\left(-\frac{x_m^2+z^2}{D^2}\right) - \exp\left(-\frac{x_p^2+z^2}{D^2}\right)\right\}\left[\mathrm{erf}\left(\frac{y_p}{D}\right) - \mathrm{erf}\left(\frac{y_m}{D}\right)\right].
 $$ (eq:Eyloop)
 
 {eq}`eq:Exloop` and {eq}`eq:Eyloop` are the expressions for the electric field anywhere in space and for all times after the source has been switched on. They show that $E_x=0$ in the $(x,z)$-plane at $y=0$ because then $y_p^2=y_m^2$ and $E_y=0$ in the $(y,z)$-plane at $x=0$ because then $x_p^2=x_m^2$. That makes sense because the contributions from the two line-segments in the $x$-direction have equal distance to all the points in the $(x,z)$ plane but have opposite currents. The equations also show that $E_y$ is the strongest in the $(x,z)$-plane at $y=0$ because then $y_p=-y_m$ and the error functions add up, and similarly $E_x$ is strongest in the $(y,z)$-plane at $x=0$ because then $x_p=-x_m$. The electric field is computed for a square loop with 100 m long segments placed in a conductive medium with $\sigma=0.3$ S/m. The result can be seen in {numref}`fig-et12xz` where $E_y$ is shown for two time instances in the $(x,z)$-plane at $y=0$, which is the cross-section right through the middle of the loop, such that the wire segments between points 4 and 1 and points 2 and 3 are perpendicular to the plane of the graph. The figure shows $E_y$ as a function of $x$ and $z$ at $t=0.03$ ms (left plot) and $t=5$ ms (right plot). It is shown only for positive depth values, but the field is symmetric in $z$ and the plot can be mirrored to add the picture of negative $z$. It can be seen that at 0.03 ms, the field is very strong and concentrated around the wires of the loop, whereas at 5 ms, the field is three orders of magnitude weaker and it almost occupies all the plane where the plot is made. Hence, we see that the maximum value of the electric field moves away from the loop wire segments and the width of the Gaussian in space widens with increasing time.
@@ -405,11 +405,11 @@ $$ (eq:Eyloop)
 The electric field $E_y$ generated by a unit step switch-on current in the $(x,z)$-plane at $y=0$ m, for times $0.03$ ms (left plot) and $5$ ms (right plot).
 ```
 
-At very early times and not too far away from the loop in the $z$-direction, the electric field shows the shape of the loop but the vector points in the opposite direction of the current vector in the loop. This can be seen in {numref}`fig-et12xy` where the magnitude of the electric field vector is shown in colour in the $(x,y)$-plane at $z=10$ m, at $t=0.03$ ms (left plot) and $t=5$ ms (right plot) and the white arrows show the direction of the electric field. At $t=0.03$ ms, the left plot shows that far away from the line segments, the electric field is fully aligned with but points in opposite direction as the current direction in the line segments, but the amplitude is very small. Curvature of the field lines going around the loop can be seen closer to the loop and is again almost fully aligned with the line segments just above them, but also here the field points in the opposite direction as the source current. At the later time of 5 ms, the right plot shows that the shape of the source loop is lost and the electric field is almost circular inside and around the loop. Notice also that the field strength has decreased by six orders of magnitude, which seems to suggest that the decay is faster in the $(x,y)$-plane for fixed depth than in the $(x,z)$-plane for fixed $y$. This will be reduced when larger depth values are chosen, because then the influence of the horizontal distances decreases.
+At very early times and not too far away from the loop in the $z$-direction, the electric field shows the shape of the loop but the vector points in the opposite direction of the current vector in the loop. This can be seen in {numref}`fig-et12xy` where the magnitude of the electric field vector is shown in colour in the $(x,y)$-plane at $z=10$ m, at $t=0.03$ ms (left plot) and $t=5$ ms (right plot) and the white arrows show the direction of the electric field. At $t=0.03$ ms, the left plot shows that far away from the line segments, the electric field is fully aligned with but points in opposite direction as the current direction in the line segments, but the amplitude is very small. Curvature of the field lines going around the loop can be seen closer to the loop and is again almost fully aligned with the line segments just above them, but also here the field points in the opposite direction as the source current. At the later time of 5 ms, the right plot shows that the shape of the source loop is lost and the electric field is almost circular inside and around the loop. Notice also that the field strength has decreased by three orders of magnitude, the same factor as in the $(x,z)$-plane. Once the diffusion distance $D$ exceeds the size of the loop, the error function bracket goes as $L_y/D$ and the difference of exponentials as $2xL_x/D^2$, so away from the symmetry axes the field falls off as $t^{-5/2}$ at every position. The rate is a property of the diffusion, not of the direction we look in.
 
 ```{figure} figures/Eyt_xy.png
 :name: fig-et12xy
 :width: 100%
 
-The electric field $E_y$ generated by a unit step switch-on current in the $(x,y)$-plane at $z=10$ m, for times $0.03$ ms (left plot) and $5$ ms (right plot).
+The magnitude $|\boldsymbol E|$ of the electric field generated by a unit step switch-on current in the $(x,y)$-plane at $z=10$ m, for times $0.03$ ms (left plot) and $5$ ms (right plot). The white arrows give the direction of the field.
 ```