diff --git a/superconductivity_assignment3_kvkempen.tex b/superconductivity_assignment3_kvkempen.tex
index 1bec9d5..f8546b3 100755
--- a/superconductivity_assignment3_kvkempen.tex
+++ b/superconductivity_assignment3_kvkempen.tex
@@ -57,18 +57,19 @@ $\kappa = \frac{\lambda}{\xi} = 34 > \frac{1}{\sqrt{2}}$,
 which means we are indeed dealing with a type-II superconductor.
 As $B_{c1} < B_E < B_{c2}$, the cylinder is in the vortex state.
 From the previous set of assignments, we know what the currents in the cylinder look like.
+From free energy considerations, we have found in lecture 4 that for type-II superconductors, it is favorable to allow flux quanta inside the superconductor in this vortex state.
+In this derivation, the contribution of one flux quantum is considered, but the consideration holds for many vortices, until they start to interact and repel eachother.
+At that point, the vortex-vortex interaction orders the vortices in a lattice.
+When the vortex cores start to overlap, there are no superconducting regions left, thus the material enters the normal conducting state.\footnote{I wanted to paint a complete picture althought it is not needed to answer the question.}
+Minimizing the free energy over the flux shows the energy is lowered for determined thresholds $B_{c1} < B_E < B_{c2}$.
 
-The average field inside the cylinder is gives as
+Let's start with the result from said free energy considerations.
+The average field inside the cylinder is given by the following self-consisting equation as
 \[
-	\langle \vec{B} \rangle = \frac{1}{V_{\text{cylinder}}} \int_{\text{cylinder}} \vec{B}(\vec{r}) d\vec{r} .
+	B = B_E - \frac{\phi_0}{8\pi\lambda^2}\ln{\frac{\phi_0}{4\exp{(1)}\xi^2B}}.
 \]
-
-To determine this $\vec{B}$ inside the material, we first need to know how many vortices there are.
-We assume that every vortex lets through only one flux quantum $\Phi_0$,
-and that the vortices will arange themselves as far as possible from eachother.
-If their distance then is large enough to assume there is no overlap between regions of finite $\vec{B}$ around them,
-we can calculate the average field by just summing over the quanta and lastly over the field that penetrates the material in the outside of the cylinder.
-For this latter calculation, we can use the field for a type-I superconductor.
+Plugging in the values for \ce{Nb3Sn}, $B_E = \SI{1}{\tesla}$, and $\phi_0 = \SI{2.0678}{\weber}$, $B$ is found as $B = \SI{.986}{\tesla} \approx B_E$ by intersection.
+%https://www.wolframalpha.com/input?i=B+%3D+1+-+%282.0678*10%5E-15%29%2F%288*pi*%28124*10%5E-9%29%5E2%29+*+ln%282.0678*10%5E-15%2F%284*exp%281%29*%283.65*10%5E-9%29%5E2+*+B%29%29
 
 \section{Superconducting wire}