Which among the following options give the correct relationship between the edge

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Question -

Which among the following options give the correct relationship between the edge lengths ‘a’ and the radius of a sphere ‘r’ in a body-centered cubic structure.

Options:

A .

a = \frac{\sqrt{3}}{4} r

B .

r = \frac{a}{2 \sqrt{2}}

C .

r = \frac{\sqrt{3}}{4} a

D .

r = \frac{a}{2}

Answer: Option C
:
C

In $△ E F D$ ,
b = a + a = 2a
Now in $△ A F D$
$c^{2} = a^{2} + b^{2} = a^{2} + 2 a^{2} = 3 a^{2}$
$\to c = \sqrt{3 a}$
The length of the body diagonal 'c' is equal to 4r, where r is the radius of the sphere (atom), as the three spheres along the diagonal touch each other.
Therefore, $\sqrt{3 a} = 4 r$
$a = \frac{4 r}{\sqrt{3}}$
$\to r = \frac{\sqrt{3}}{4} a$

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