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 inthese systems will be respectively,

Options

A. `(1)/(2)a:(sqrt3)/(4)a:(1)/(2sqrt2)a`
B. `(1)/(2)a:sqrt3a:(1)/(sqrt2)a`
C. `(1)/(2)a:(sqrt3)/(2)a:(sqrt2)/(2)a`
D. `1a:sqrt3a:sqrt2a`

Explanation

To solve the problem of finding the ratio of the radii of spheres in simple cubic, body-centered cubic, and face-centered cubic systems, we will follow these steps:

### Step 1: Understand the relationship between edge length and radius in different cubic systems.

1. **Simple Cubic (SC)**:
- The edge length \( a \) is related to the radius \( r \) of the sphere by the formula:
\[ a = 2r \quad \Rightarrow \quad r = \frac{a}{2} \]

2. **Body-Centered Cubic (BCC)**:
- The edge length \( a \) is related to the radius \( R \) of the sphere by the formula:
\[ \sqrt{3}a = 4R \quad \Rightarrow \quad R = \frac{\sqrt{3}}{4}a \]

3. **Face-Centered Cubic (FCC)**:
- The edge length \( a \) is related to the radius \( r \) of the sphere by the formula:
\[ \sqrt{2}a = 4r \quad \Rightarrow \quad r = \frac{\sqrt{2}}{4}a = \frac{1}{2\sqrt{2}}a \]

### Step 2: Write down the expressions for the radii.

From the above relationships, we have:
- For Simple Cubic: \( r_{SC} = \frac{a}{2} \)
- For Body-Centered Cubic: \( r_{BCC} = \frac{\sqrt{3}}{4}a \)
- For Face-Centered Cubic: \( r_{FCC} = \frac{1}{2\sqrt{2}}a \)

### Step 3: Find the ratio of the radii.

Now, we can find the ratio of the radii:
\[ \text{Ratio} = r_{SC} : r_{BCC} : r_{FCC} = \frac{a}{2} : \frac{\sqrt{3}}{4}a : \frac{1}{2\sqrt{2}}a \]

### Step 4: Simplify the ratio.

To simplify, we can divide each term by \( a \):
\[ \text{Ratio} = \frac{1}{2} : \frac{\sqrt{3}}{4} : \frac{1}{2\sqrt{2}} \]

Now, we can express each term with a common denominator, which is 4:
- \( \frac{1}{2} = \frac{2}{4} \)
- \( \frac{\sqrt{3}}{4} = \frac{\sqrt{3}}{4} \)
- \( \frac{1}{2\sqrt{2}} = \frac{2}{4\sqrt{2}} \)

Thus, the ratio becomes:
\[ \text{Ratio} = 2 : \sqrt{3} : \frac{2}{\sqrt{2}} \]

### Step 5: Finalize the ratio.

To express the last term in a simpler form, we can multiply through by \( \sqrt{2} \):
\[ \text{Ratio} = 2\sqrt{2} : \sqrt{3} : 2 \]

### Conclusion:

The final ratio of the radii of the spheres in the simple cubic, body-centered cubic, and face-centered cubic systems is:
\[ \text{Ratio} = 1 : \frac{\sqrt{3}}{2} : \frac{1}{2\sqrt{2}} \]