If e is the electronic charge, c is the speed of light in free space and h is Planck's constant, the quantity `(1)/(4pi epsilon_(0)) (|e|^(2))/(hc)` has dimensions of :

Options

A. `[M^(0)L^(0)T^(0)]`
B. `[L C^(-1)]`
C. `[MLT^(-1)]`
D. `[MLT^(0)]`

Answer

To find the dimensions of the quantity \(\frac{1}{4\pi \epsilon_0} \frac{|e|^2}{hc}\), we will break it down into its components and analyze the dimensions of each part step by step.

### Step 1: Identify the components

The expression consists of three main components:
1. \(\epsilon_0\) (the permittivity of free space)
2. \(e\) (the electronic charge)
3. \(h\) (Planck's constant)
4. \(c\) (the speed of light)

### Step 2: Write down the dimensions of each component

1. **Permittivity of free space (\(\epsilon_0\))**:
The dimension of \(\epsilon_0\) can be derived from the relation of electric force:
\[ F = \frac{1}{4\pi \epsilon_0} \frac{q_1 q_2}{r^2} \]
Rearranging gives:
\[ \epsilon_0 = \frac{q_1 q_2}{F \cdot r^2} \]
The dimensions of force \(F\) are \([M L T^{-2}]\), charge \(q\) is \([I T]\), and distance \(r\) is \([L]\). Thus, the dimensions of \(\epsilon_0\) are:
\[ [\epsilon_0] = \frac{[I T]^2}{[M L T^{-2}] [L^2]} = \frac{I^2 T^2}{M L^3} \]

2. **Electronic charge (\(e\))**:
The dimension of charge \(e\) is:
\[ [e] = [I T] \]

3. **Planck's constant (\(h\))**:
The dimension of Planck's constant \(h\) is:
\[ [h] = [E][T] = [M L^2 T^{-2}][T] = [M L^2 T^{-1}] \]

4. **Speed of light (\(c\))**:
The dimension of the speed of light \(c\) is:
\[ [c] = [L T^{-1}] \]

### Step 3: Substitute the dimensions into the expression

Now, we can substitute these dimensions into the expression:
\[ \frac{1}{4\pi \epsilon_0} \frac{|e|^2}{hc} \]

Ignoring the constant \(\frac{1}{4\pi}\) as it has no dimensions, we have:
\[ \text{Dimensions of } \frac{|e|^2}{hc} = \frac{[e]^2}{[h][c]} = \frac{(I T)^2}{(M L^2 T^{-1})(L T^{-1})} \]

### Step 4: Simplify the dimensions

Substituting the dimensions:
\[ = \frac{I^2 T^2}{M L^2 T^{-1} \cdot L T^{-1}} = \frac{I^2 T^2}{M L^3 T^{-2}} = \frac{I^2 T^4}{M L^3} \]

### Step 5: Combine with \(\epsilon_0\)

Now, we need to include the dimensions of \(\epsilon_0\):
\[ \text{Dimensions of } \frac{1}{\epsilon_0} = \frac{M L^3}{I^2 T^2} \]

Thus, the overall dimensions become:
\[ \text{Dimensions of } \frac{1}{\epsilon_0} \cdot \frac{|e|^2}{hc} = \left(\frac{M L^3}{I^2 T^2}\right) \cdot \left(\frac{I^2 T^4}{M L^3}\right) \]

### Step 6: Final simplification

This simplifies to:
\[ = \frac{M L^3}{I^2 T^2} \cdot \frac{I^2 T^4}{M L^3} = 1 \cdot T^2 = T^2 \]

### Conclusion

The dimensions of the quantity \(\frac{1}{4\pi \epsilon_0} \frac{|e|^2}{hc}\) are:
\[ [T^2] \]