If `e, in_0, h and c` respectively represent electric, charge, permittivity of free space, Planck's constant and speed of light then `(e^2)/(in_0 hc)` has the dimensions of
a) angle
b) relative density
c) strain
d) current

Options

A. a & b are correct
B. d & c are correct
C. a,b & c are correct
D. a,b,c & d are correct

Answer

To solve the problem, we need to find the dimensions of the expression \(\frac{e^2}{\epsilon_0 hc}\), where:
- \(e\) is the electric charge,
- \(\epsilon_0\) is the permittivity of free space,
- \(h\) is Planck's constant,
- \(c\) is the speed of light.

### Step 1: Write down the dimensions of each quantity

1. **Electric Charge (\(e\))**: The dimension of electric charge is given by:
\[ [e] = [I][T] = I^1 T^1 \]

2. **Permittivity of Free Space (\(\epsilon_0\))**: The dimension of permittivity is:
\[ [\epsilon_0] = \frac{[M]^{-1}[L]^{-3}[T]^4}{[I]^2} = M^{-1} L^{-3} T^4 I^2 \]

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

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

### Step 2: Substitute the dimensions into the expression

Now we substitute these dimensions into the expression \(\frac{e^2}{\epsilon_0 hc}\):

\[ [e^2] = (I^1 T^1)^2 = I^2 T^2 \]

Now, substituting into the expression:

\[ \frac{e^2}{\epsilon_0 hc} = \frac{I^2 T^2}{(M^{-1} L^{-3} T^4 I^2)(M^1 L^2 T^{-1})(L^1 T^{-1})} \]

### Step 3: Simplify the expression

Now we simplify the denominator:

\[ \epsilon_0 hc = (M^{-1} L^{-3} T^4 I^2)(M^1 L^2 T^{-1})(L^1 T^{-1}) = M^0 L^{-3 + 2 + 1} T^{4 - 1 - 1} I^2 = M^0 L^{-1} T^2 I^2 \]

Thus, the expression becomes:

\[ \frac{I^2 T^2}{M^0 L^{-1} T^2 I^2} = \frac{I^2 T^2}{L^{-1} T^2 I^2} = L^1 \]

### Step 4: Final Result

The dimensions of \(\frac{e^2}{\epsilon_0 hc}\) simplify to:

\[ [L] \]

This indicates that the expression has the dimensions of length, which is not directly listed in the options. However, we can analyze the options:

- a) angle
- b) relative density
- c) strain
- d) current

Since the dimension of length does not match any of the options, we can conclude that the expression has no specific dimension from the given options.

### Conclusion

The answer is that the expression \(\frac{e^2}{\epsilon_0 hc}\) does not correspond to any of the provided options, but if we consider the context of dimensionless quantities, the closest match would be option (a) angle, as angles are dimensionless.