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Q.
The packing efficiency of the two-dimensional square unit cell shown below is
IIT JEEIIT JEE 2010The Solid State
Solution:
Contribution of circle from corner of square $=\frac{1}{4}$
$\Rightarrow$ Effective number of circle per square
$=\frac{1}{4} \times 4+1$ (at centre) $=2$
$\Rightarrow$ Area occupied by circle $=2 \pi r^{2}, r=$ radius.
Also, diagonal of square $4 r=\sqrt{2} L$, where $L=$ side of square.
$\Rightarrow$ Packing fraction $=\frac{\text { Area occupied by circles }}{\text { Area of square }}$
$=\frac{2 \pi r^{2}}{L^{2}}=\frac{2 \pi r^{2}}{8 r^{2}}=\frac{\pi}{4}=0.785$
$\Rightarrow \%$ packing efficiency $=78.5 \%$.
