Consider three quantities (`x= E/B, y= (sqrt(1/((mu_0)(epsilon_0)))) and z=(1/CR)'. Here, l is the length of a wire, C is a capacitance and R is a resistance. All other symbols have standard meanings.

Options

A. x,y have the same dimensions.
B. y,z have the same dimensions.
C. z,x have the same dimensions.
D. none of the three pairs have the same dimensions.

Answer

To solve the problem, we need to analyze the dimensions of the three quantities given: \( x = \frac{E}{B} \), \( y = \sqrt{\frac{1}{\mu_0 \epsilon_0}} \), and \( z = \frac{l}{CR} \).

### Step-by-Step Solution:

1. **Determine the dimensions of \( x = \frac{E}{B} \)**:
- \( E \) is the electric field, which has the dimension of \( \text{MLT}^{-3}\text{T}^2 = \text{ML}^2\text{T}^{-3} \) (derived from \( F = qE \)).
- \( B \) is the magnetic field, which has the dimension of \( \text{MLT}^{-2}\text{T}^{-1} = \text{ML}^2\text{T}^{-2} \) (derived from \( F = qvB \)).
- Therefore, the dimension of \( x \) is:
\[ [x] = \frac{[E]}{[B]} = \frac{\text{ML}^2\text{T}^{-3}}{\text{ML}^2\text{T}^{-2}} = \text{T}^{-1} = \text{LT}^{-1} \]

2. **Determine the dimensions of \( y = \sqrt{\frac{1}{\mu_0 \epsilon_0}} \)**:
- The permeability \( \mu_0 \) has dimensions of \( \text{MLT}^{-2}\text{A}^{-2} \).
- The permittivity \( \epsilon_0 \) has dimensions of \( \text{ML}^{-3}\text{T}^4\text{A}^2 \).
- Therefore, the dimension of \( \mu_0 \epsilon_0 \) is:
\[ [\mu_0 \epsilon_0] = [\mu_0][\epsilon_0] = (\text{MLT}^{-2}\text{A}^{-2})(\text{ML}^{-3}\text{T}^4\text{A}^2) = \text{M}^2\text{L}^{-2}\text{T}^2 \]
- Thus, the dimension of \( y \) is:
\[ [y] = \sqrt{\frac{1}{[\mu_0 \epsilon_0]}} = \sqrt{\frac{1}{\text{M}^2\text{L}^{-2}\text{T}^2}} = \text{LT}^{-1} \]

3. **Determine the dimensions of \( z = \frac{l}{CR} \)**:
- The length \( l \) has the dimension of \( L \).
- The capacitance \( C \) has the dimension of \( \frac{\text{A}^2\text{T}^4}{\text{ML}} \).
- The resistance \( R \) has the dimension of \( \frac{\text{ML}^2}{\text{A}^2\text{T}^3} \).
- Therefore, the dimension of \( CR \) is:
\[ [CR] = [C][R] = \left(\frac{\text{A}^2\text{T}^4}{\text{ML}}\right)\left(\frac{\text{ML}^2}{\text{A}^2\text{T}^3}\right) = \text{LT} \]
- Thus, the dimension of \( z \) is:
\[ [z] = \frac{[l]}{[CR]} = \frac{L}{\text{LT}} = \text{T}^{-1} = \text{LT}^{-1} \]

4. **Conclusion**:
- We find that all three quantities \( x \), \( y \), and \( z \) have the same dimensions:
\[ [x] = [y] = [z] = \text{LT}^{-1} \]

### Final Answer:
- The correct options are: **A**, **B**, and **C** (all pairs have the same dimensions).