Which of the following pairs have the same dimensions?

Options

A. ` h//e` and magnetic flux
B. ` h//e` and electric flux
C. Electric fux and `q//epsilon_(0)`
D. Electric flux and `mu_(0)I`

Explanation

To determine which pairs have the same dimensions, we will analyze the dimensions of the given quantities step by step.

### Step 1: Analyzing H/E
1. **Identify H**: The quantity H (magnetic field strength) has units of Joule per second (J/s).
- Joule (J) = kg·m²/s², so:
- H = J/s = (kg·m²/s²) / s = kg·m²/s³ = [M L² T⁻³]

2. **Identify E**: The quantity E (electric field) has units of voltage (V) per unit of charge (C).
- Voltage (V) = J/C = (kg·m²/s²) / (A·s) = kg·m²/(A·s³), so:
- E = V/C = (kg·m²/(A·s³)) / (A·s) = kg·m²/(A²·s⁴) = [M L² T⁻⁴ A⁻²]

3. **Calculate H/E**:
- H/E = [M L² T⁻³] / [M L² T⁻⁴ A⁻²] = [A² T] = [M L² T⁻² A⁻¹]

### Step 2: Analyzing Magnetic Flux
1. **Magnetic Flux (Φ)**: Magnetic flux is defined as the product of the magnetic field (B) and the area (A).
- The dimension of magnetic flux (Φ) is given by:
- Φ = B·A
- We know B = F/(I·L), where F = M L T⁻², I = A, and L = L.
- Thus, B = [M L T⁻²] / [A·L] = [M T⁻² A⁻¹].

2. **Area (A)**: The dimension of area is L².

3. **Combine dimensions**:
- Φ = [M T⁻² A⁻¹]·[L²] = [M L² T⁻² A⁻¹].

### Step 3: Compare H/E and Magnetic Flux
- From the calculations:
- H/E = [M L² T⁻² A⁻¹]
- Magnetic Flux = [M L² T⁻² A⁻¹]

Thus, H/E and Magnetic Flux have the same dimensions.

### Step 4: Analyzing Electric Flux
1. **Electric Flux (Φ_e)**: Electric flux is defined as the charge (Q) divided by the permittivity (ε₀).
- Φ_e = Q / ε₀.

2. **Charge (Q)**: The dimension of charge is given by:
- Q = [A·T].

3. **Permittivity (ε₀)**: The dimension of permittivity is:
- ε₀ = [M⁻¹ L⁻³ T⁴ A²].

4. **Combine dimensions**:
- Φ_e = [A·T] / [M⁻¹ L⁻³ T⁴ A²] = [M L³ T⁻³ A⁻¹].

### Step 5: Final Comparison
- Electric Flux = [M L³ T⁻³ A⁻¹].
- This is different from the dimensions of H/E and Magnetic Flux.

### Conclusion
The pairs that have the same dimensions are:
1. H/E and Magnetic Flux.