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If ' $a$ ' stands for the edge length of the cubic systems : simple cubic, body centred cubic and face centred cubic, then the ratio of radii of the spheres in these systems will be respectively,
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Verified Answer
The correct answer is:
$\frac{1}{2} a : \frac{\sqrt{3}}{4} a: \frac{1}{2 \sqrt{2}} a$
For simple cubic,
$\begin{aligned}
\therefore & a & =2 r \\
\therefore & r & =\frac{a}{2}
\end{aligned}$
For body centred cubic,
$\begin{aligned}
& a=\frac{4 r}{\sqrt{3}} \\
& r=\frac{\sqrt{3} a}{4}
\end{aligned}$
For face centred cubic,
$\begin{aligned}
& a=2 \sqrt{2} r \\
& r=\frac{a}{2 \sqrt{2}}
\end{aligned}$
Hence, the ratio of radii in simple cubic, body centred cubic and face centred cubic is $\frac{a}{2}: \frac{\sqrt{3} a}{4}: \frac{a}{2 \sqrt{2}}$
$\begin{aligned}
\therefore & a & =2 r \\
\therefore & r & =\frac{a}{2}
\end{aligned}$
For body centred cubic,
$\begin{aligned}
& a=\frac{4 r}{\sqrt{3}} \\
& r=\frac{\sqrt{3} a}{4}
\end{aligned}$
For face centred cubic,
$\begin{aligned}
& a=2 \sqrt{2} r \\
& r=\frac{a}{2 \sqrt{2}}
\end{aligned}$
Hence, the ratio of radii in simple cubic, body centred cubic and face centred cubic is $\frac{a}{2}: \frac{\sqrt{3} a}{4}: \frac{a}{2 \sqrt{2}}$
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