A metal sample carrying a current along x axis with density is subjected to a magnetic field `B_(z)` (along z - axis). The electric field `E_(y)` developed along Y-axis is directly proportional to `J_(x)` as well as `B_(z)`. The constant of proportionality has SI unit.

A. `m^2//A`
B. `m^3//As`
C. `m^2//As`
D. `As//m^3`

Options

A. `m^2//A`
B. `m^3//As`
C. `m^2//As`
D. `As//m^3`

Answer

To solve the given problem, we need to analyze the relationship between the electric field \( E_y \), the current density \( J_x \), and the magnetic field \( B_z \).

### Step-by-Step Solution:

1. **Understanding the Relationship**:
We are given that the electric field \( E_y \) is directly proportional to the current density \( J_x \) and the magnetic field \( B_z \). This can be expressed mathematically as:
\[ E_y \propto J_x \cdot B_z \]

2. **Introducing a Proportionality Constant**:
To convert the proportionality into an equation, we introduce a constant of proportionality \( k \):
\[ E_y = k \cdot J_x \cdot B_z \]

3. **Rearranging for the Constant \( k \)**:
We can rearrange the equation to solve for \( k \):
\[ k = \frac{E_y}{J_x \cdot B_z} \]

4. **Identifying the Units**:
Now, we need to determine the units of \( k \). We know the following units:
- The unit of electric field \( E_y \) is volts per meter (V/m).
- The unit of current density \( J_x \) is amperes per square meter (A/m²).
- The unit of magnetic field \( B_z \) is teslas (T).

5. **Converting Units**:
We can express the units in terms of base SI units:
- \( 1 \, \text{V} = 1 \, \text{kg} \cdot \text{m}^2 / (\text{s}^3 \cdot \text{A}) \)
- \( 1 \, \text{T} = 1 \, \text{kg} / (\text{s}^2 \cdot \text{A}) \)

Therefore, the units of \( E_y \) can be expressed as:
\[ [E_y] = \frac{\text{kg} \cdot \text{m}^2}{\text{s}^3 \cdot \text{A} \cdot \text{m}} = \frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{A}} \]

6. **Calculating the Units of \( k \)**:
Now substituting the units into the expression for \( k \):
\[ k = \frac{[E_y]}{[J_x] \cdot [B_z]} = \frac{\frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{A}}}{\left(\frac{\text{A}}{\text{m}^2}\right) \cdot \left(\frac{\text{kg}}{\text{s}^2 \cdot \text{A}}\right)} \]

Simplifying this gives us:
\[ k = \frac{\frac{\text{kg} \cdot \text{m}}{\text{s}^3 \cdot \text{A}}}{\frac{\text{kg} \cdot \text{m}^2}{\text{s}^2 \cdot \text{A}^2}} = \frac{\text{m}}{\text{s} \cdot \text{A}} = \text{m}^2/\text{As} \]

7. **Final Result**:
The constant of proportionality \( k \) has the SI unit of \( \text{m}^2/\text{As} \).

### Conclusion:
Thus, the correct answer is:
**C. \( \text{m}^2/\text{As} \)**