The electric field in a region is given by , `E=(A)/(x^(2))hati+Byhatj+Cz^(2)hatk`
The SI units of A ,B and C are , respectively

Options

A. `(Nm^(3))/(C),Vm^(2),(N)/(m^(2)C)`
B. `Vm^2 ,Vm, (N)/(m ^2 C)`
C. `V//m^(2),V//m,(NC)/(m^(2))`
D. `V//m^2 (Nm^1 )/( c ) (N )/(C )`

Explanation

To determine the SI units of A, B, and C in the given electric field expression \( E = \frac{A}{x^2} \hat{i} + B y \hat{j} + C z^2 \hat{k} \), we will follow these steps:

### Step 1: Identify the SI unit of the electric field
The electric field \( E \) is defined as force per unit charge. The SI unit of electric field is:
\[ \text{SI unit of } E = \frac{\text{Force}}{\text{Charge}} = \frac{\text{Newton}}{\text{Coulomb}} = \text{N/C} \]

### Step 2: Analyze the first term \( \frac{A}{x^2} \)
For the term \( \frac{A}{x^2} \) to have the same unit as the electric field, we set:
\[ \frac{A}{x^2} = E \]
Thus, the SI unit of \( \frac{A}{x^2} \) must also be \( \text{N/C} \).

Rearranging gives:
\[ A = E \cdot x^2 \]
Substituting the unit of \( E \):
\[ A = \text{N/C} \cdot \text{m}^2 = \text{N} \cdot \text{m}^2/\text{C} \]

### Step 3: Analyze the second term \( By \)
For the term \( By \) to have the same unit as the electric field, we set:
\[ By = E \]
Thus, the SI unit of \( By \) must also be \( \text{N/C} \).

Rearranging gives:
\[ B = \frac{E}{y} \]
Substituting the unit of \( E \):
\[ B = \frac{\text{N/C}}{\text{m}} = \text{N}/(\text{C} \cdot \text{m}) \]

### Step 4: Analyze the third term \( Cz^2 \)
For the term \( Cz^2 \) to have the same unit as the electric field, we set:
\[ Cz^2 = E \]
Thus, the SI unit of \( Cz^2 \) must also be \( \text{N/C} \).

Rearranging gives:
\[ C = \frac{E}{z^2} \]
Substituting the unit of \( E \):
\[ C = \frac{\text{N/C}}{\text{m}^2} = \text{N}/(\text{C} \cdot \text{m}^2) \]

### Summary of SI Units
- The SI unit of \( A \) is \( \text{N} \cdot \text{m}^2/\text{C} \)
- The SI unit of \( B \) is \( \text{N}/(\text{C} \cdot \text{m}) \)
- The SI unit of \( C \) is \( \text{N}/(\text{C} \cdot \text{m}^2) \)

### Final Answer
The SI units of A, B, and C are:
- \( A: \text{N} \cdot \text{m}^2/\text{C} \)
- \( B: \text{N}/(\text{C} \cdot \text{m}) \)
- \( C: \text{N}/(\text{C} \cdot \text{m}^2) \)