When a 1/2 inch diameter bar is inserted into a 1 inch diameter coil, what is the fill factor?

To determine the fill factor when a 1/2 inch diameter bar is inserted into a 1 inch diameter coil, the fill factor is calculated as the ratio of the cross-sectional area of the bar to the cross-sectional area of the coil.

First, calculate the cross-sectional area of the bar. The area ( A ) of a circle is given by the formula ( A = \pi \times r^2 ). For the bar with a diameter of 1/2 inch, the radius is 1/4 inch. Thus, the area of the bar is:

[

A_{bar} = \pi \times (1/4)^2 = \pi \times (1/16) = \frac{\pi}{16}

]

Next, calculate the cross-sectional area of the coil. The coil has a diameter of 1 inch, which means the radius is 1/2 inch. Therefore, the area of the coil is:

[

A_{coil} = \pi \times (1/2)^2 = \pi \times (1/4) = \frac{\pi}{4}

]

Now, the fill factor is the ratio of the area of the bar to the area of