question_answer

Relationship between atomic radius r and the edge length a of a body centred cubic unit cell is

A)  \[r=\frac{a}{2}\]                             

B)  \[r=\sqrt{\frac{a}{2}}\]

C)  \[r=\frac{\sqrt{3}}{4}a\]                              

D)  \[r=\frac{3a}{2}\]

Correct Answer: C

Solution :

                From the figure, In\[\Delta PQR,\] \[{{b}^{2}}={{a}^{2}}+{{a}^{2}}\] [From\[{{(AB)}^{2}}={{(AC)}^{2}}+{{(BC)}^{2}}\]] \[{{b}^{2}}=2{{a}^{2}}\]                 \[b=\sqrt{2}a\] In \[\Delta PRS,\] \[{{(4r)}^{2}}={{(SR)}^{2}}+{{(PR)}^{2}}\]                 \[={{(a)}^{2}}+{{(\sqrt{2}a)}^{2}}\]                 \[{{(4r)}^{2}}=3{{a}^{2}}\]                 \[4r=\sqrt{3}a\]                 \[r=\frac{\sqrt{3}}{4}a\]