$[$ a $\backslash$sin$(\backslash$theta$\_$n$)=$ n $\backslash$lambda $]$

where:

$($a$)$ is the width of the slit,

$(\backslash$lambda$)$ is the wavelength of light,

$($n$)$ is the order of the dark fringe,

$(\backslash$theta$\_$n$)$ is the angle corresponding to the $($n$)$th minimum.

Step $1$: Set the parameters

Given that $($a $=4\mathrm{.}3\backslash$lambda$),$ we can express the condition for the dark fringes as:

$[4\mathrm{.}3\backslash$lambda $\backslash$sin$(\backslash$theta$\_$n$)=$ n $\backslash$lambda $]$

Dividing both sides by $(\backslash$lambda$)$:

$[4\mathrm{.}3\backslash$sin$(\backslash$theta$\_$n$)=$ n $]$

Step $2$: Find the maximum $($n$)$

To find the maximum number of dark fringes, we must note that $(\backslash$sin$(\backslash$theta$\_$n$))$ can take values between $(-1)$ and $(1).$ Thus, we set:

$[4\mathrm{.}3\backslash$sin$(\backslash$theta$\_$n$)\backslash$leq $4\mathrm{.}3]$

This means:

$[$ n $\backslash$leq $4\mathrm{.}3]$

Since $($n$)$ must be a whole number, the maximum integer value for $($n$)$ is $(4).$

Step $3$: Fringes on a semicircular screen

Each dark fringe corresponds to a dark region in the diffraction pattern. For a semicircular screen, the maximum number of minima that can be observed is equal to the number of integer orders possible:

The dark fringes are located on either side of the central maximum, but since we are considering a semicircular screen, we only need the minimum values without repeating counts.

Conclusion

Thus, the maximum number of dark fringes observable on a semicircular screen surrounding the slit is:

$[\backslash$boxed$\{4\}$