Step 1: Recall the rules of significant figures.
Leading zeros appearing before the first non-zero digit are not counted as significant figures.
In the number
\[
0.00005041,
\]
the zeros before the digit \(5\) are only placeholders and are not significant.
Step 2: Identify the significant digits.
The significant digits are
\[
5,\;0,\;4,\;1
\]
The zero between \(5\) and \(4\) is significant because it occurs between non-zero digits.
Step 3: Count the significant figures.
Thus, the total number of significant figures is
\[
4
\]
Step 4: Final conclusion.
Therefore, the number \(0.00005041\ \text{J}\) contains
\[
\boxed{4}
\]
significant figures.