Consider a hollow sphere $A$ of radius $1$, containing a concentric sphere $B$ of radius $r < 1$. I am trying to calculate the view factor $F_{A \rightarrow B}$, but different approaches give me different results.

Approach 1

The apparent radius of $B$, as seen from any point of $A$, is: $\theta = \arcsin(r)$.
The solid angle occupied by $B$ from that point of view is: $\Omega = 2\pi(1 - \cos(\theta))$.
The point of $A$ emits radiation in every directions uniformly across the hemisphere, so the fraction that hits $B$ is: $\Omega / 2\pi = 1 - \cos(\arcsin(r))$.
Since this is true of all points of $A$, we have $F_{A \rightarrow B} = 1 - \cos(\arcsin(r))$

Approach 2

The areas of $A$ and $B$ are $A_A = 4\pi$ and $A_B = 4\pi r^2$, respectively.
Since B is convex, we have $F_{B \rightarrow B} = 0$ and therefore $F_{B \rightarrow A} = 1$.
By the reciprocity theorem, we have $A_A F_{A \rightarrow B} = A_B F_{B \rightarrow A}$ and therefore $F_{A \rightarrow B} = r^2$

Obviously these results are different. I assume the theorem is correct and my first approach is flawed. Where did I go wrong?


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