Find the dimension of `(1)/((4pi)epsilon_(0))(e^(2))/(hc))`

Answer

To find the dimension of the expression \((1)/((4\pi)\epsilon_{0})(e^{2})/(hc)\), we will break it down step by step.

### Step 1: Understand the components of the expression

The expression consists of the following components:
- \(\epsilon_{0}\): The permittivity of free space.
- \(e\): The charge of an electron.
- \(h\): Planck's constant.
- \(c\): 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 formula for the force between two charges:
\[ F = \frac{1}{4\pi\epsilon_{0}} \frac{q_1 q_2}{r^2} \]
Rearranging gives:
\[ \epsilon_{0} = \frac{q_1 q_2}{4\pi F r^2} \]
Hence, the dimension of \(\epsilon_{0}\) is:
\[ [\epsilon_{0}] = \frac{[q]^2}{[F][L^2]} = \frac{[I^2 T^4]}{[M L T^{-2}][L^2]} = \frac{[I^2 T^4]}{[M L^3 T^{-2}]} = [M^{-1} L^{-3} T^4 I^2] \]

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

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

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

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

Now we can substitute the dimensions into the expression:
\[ \frac{1}{(4\pi)\epsilon_{0}} \cdot \frac{e^2}{hc} \]

Substituting the dimensions:
\[ \text{Dimension of } \frac{1}{\epsilon_{0}} = [M L^3 T^{-4} I^{-2}] \]
\[ \text{Dimension of } e^2 = [I^2 T^2] \]
\[ \text{Dimension of } hc = [M L^2 T^{-1}][L T^{-1}] = [M L^3 T^{-2}] \]

### Step 4: Combine the dimensions

Now, we can combine the dimensions:
\[ \text{Dimension of the expression} = [M L^3 T^{-4} I^{-2}] \cdot [I^2 T^2] \cdot [M^{-1} L^{-3} T^{2}] \]

### Step 5: Simplify the expression

Combining these gives:
\[ = [M L^3 T^{-4} I^{-2}] \cdot [I^2 T^2] \cdot [M^{-1} L^{-3} T^{2}] \]
\[ = [M^{1-1} L^{3-3} T^{-4+2+2} I^{2-2}] = [M^0 L^0 T^0 I^0] = \text{Dimensionless} \]

### Final Answer

The dimension of the expression \(\frac{1}{(4\pi)\epsilon_{0}} \cdot \frac{e^2}{hc}\) is **dimensionless**.