Which of these is/are representing electrostatic field? (`x` and `y` are space coordinates)

Options

A. `yhat(i)+xhat(j)`
B. `sinxhat(i)`
C. `xhat(i)+yhat(j)`
D. `x^(2)hat(i)+yhat(j)`

Explanation

To determine which of the given fields represent electrostatic fields, we need to apply the condition for an electrostatic field. The condition states that for a field represented as \( \mathbf{E} = E_x \hat{i} + E_y \hat{j} \), the following must hold true:

\[ \frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x} \]

We will analyze each option step by step.

### Step 1: Analyze Option A
Given field: \( \mathbf{E} = y \hat{i} + x \hat{j} \)

- \( E_x = y \)
- \( E_y = x \)

Now, calculate the derivatives:
- \( \frac{\partial E_x}{\partial y} = \frac{\partial y}{\partial y} = 1 \)
- \( \frac{\partial E_y}{\partial x} = \frac{\partial x}{\partial x} = 1 \)

Since both derivatives are equal:
\[ \frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x} \Rightarrow 1 = 1 \]
**Conclusion**: Option A represents an electrostatic field.

### Step 2: Analyze Option B
Given field: \( \mathbf{E} = \sin(x) \hat{i} + 0 \hat{j} \)

- \( E_x = \sin(x) \)
- \( E_y = 0 \)

Now, calculate the derivatives:
- \( \frac{\partial E_x}{\partial y} = \frac{\partial \sin(x)}{\partial y} = 0 \)
- \( \frac{\partial E_y}{\partial x} = \frac{\partial 0}{\partial x} = 0 \)

Since both derivatives are equal:
\[ \frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x} \Rightarrow 0 = 0 \]
**Conclusion**: Option B represents an electrostatic field.

### Step 3: Analyze Option C
Given field: \( \mathbf{E} = x \hat{i} + y \hat{j} \)

- \( E_x = x \)
- \( E_y = y \)

Now, calculate the derivatives:
- \( \frac{\partial E_x}{\partial y} = \frac{\partial x}{\partial y} = 0 \)
- \( \frac{\partial E_y}{\partial x} = \frac{\partial y}{\partial x} = 0 \)

Since both derivatives are equal:
\[ \frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x} \Rightarrow 0 = 0 \]
**Conclusion**: Option C represents an electrostatic field.

### Step 4: Analyze Option D
Given field: \( \mathbf{E} = x^2 \hat{i} + y \hat{j} \)

- \( E_x = x^2 \)
- \( E_y = y \)

Now, calculate the derivatives:
- \( \frac{\partial E_x}{\partial y} = \frac{\partial x^2}{\partial y} = 0 \)
- \( \frac{\partial E_y}{\partial x} = \frac{\partial y}{\partial x} = 0 \)

Since both derivatives are equal:
\[ \frac{\partial E_x}{\partial y} = \frac{\partial E_y}{\partial x} \Rightarrow 0 = 0 \]
**Conclusion**: Option D represents an electrostatic field.

### Final Conclusion
All four options A, B, C, and D represent electrostatic fields.

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