1. The problem involves understanding a circular arc centered at the fraction $\frac{1}{2}$ with a radius given by the fraction $\frac{2}{4}$.\n\n2. The center of the circle is at $\left(\frac{1}{2}, 0\right)$ assuming the arc lies on the x-axis for simplicity.\n\n3. The radius is $\frac{2}{4} = \frac{1}{2}$.\n\n4. The equation of a circle with center $(h,k)$ and radius $r$ is $$ (x - h)^2 + (y - k)^2 = r^2 $$\n\n5. Substituting $h = \frac{1}{2}$, $k = 0$, and $r = \frac{1}{2}$, we get $$ \left(x - \frac{1}{2}\right)^2 + y^2 = \left(\frac{1}{2}\right)^2 $$\n\n6. Simplifying the radius squared: $$ \left(\frac{1}{2}\right)^2 = \frac{1}{4} $$\n\n7. Therefore, the equation of the circle is $$ \left(x - \frac{1}{2}\right)^2 + y^2 = \frac{1}{4} $$\n\nThis equation describes the circular arc centered at $\frac{1}{2}$ with radius $\frac{1}{2}$.