The gravitational field in a region is given by `vecE=(yhati+ xhatj)` N/kg, where x and y are in metres. The equipotential lines are plotted as

Options

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C.
D.

Answer

To find the equipotential lines for the given gravitational field \(\vec{E} = (y \hat{i} + x \hat{j})\) N/kg, we need to understand that equipotential lines are the lines along which the potential energy is constant.

### Step-by-Step Solution:

1. **Understand the Gravitational Field**: The gravitational field is given as \(\vec{E} = (y \hat{i} + x \hat{j})\). This means that the gravitational field has components in both the x and y directions, which depend on the coordinates \(x\) and \(y\).

2. **Relate Gravitational Field to Potential**: The gravitational potential \(V\) is related to the gravitational field \(\vec{E}\) by the equation:
\[ \vec{E} = -\nabla V \]
This means that the components of the gravitational field can be expressed as:
\[ E_x = -\frac{\partial V}{\partial x} \quad \text{and} \quad E_y = -\frac{\partial V}{\partial y} \]

3. **Set Up the Equations**: From the given field, we have:
\[ E_x = y \quad \text{and} \quad E_y = x \]
Therefore, we can write:
\[ -\frac{\partial V}{\partial x} = y \quad \text{(1)} \]
\[ -\frac{\partial V}{\partial y} = x \quad \text{(2)} \]

4. **Integrate Equation (1)**: To find \(V\), we can integrate equation (1) with respect to \(x\):
\[ V = -\int y \, dx = -yx + f(y) \]
where \(f(y)\) is an arbitrary function of \(y\).

5. **Differentiate with respect to \(y\)**: Now, we differentiate \(V\) with respect to \(y\) and set it equal to equation (2):
\[ \frac{\partial V}{\partial y} = -x + f'(y) \]
Setting this equal to \(-x\) gives:
\[ -x + f'(y) = -x \]
This implies that \(f'(y) = 0\), so \(f(y)\) is a constant. Thus, we can write:
\[ V = -yx + C \]
where \(C\) is a constant.

6. **Equipotential Lines**: The equipotential lines are given by setting \(V\) to a constant value \(V_0\):
\[ -yx + C = V_0 \]
Rearranging gives:
\[ yx = C - V_0 \]
This equation represents hyperbolas in the \(xy\)-plane.

### Final Result:
The equipotential lines are given by the equation:
\[ yx = \text{constant} \]