The electric potential at a point `(x,y)` in the x-y plane is given by `V=-kxy`. The field intentisy at a distance r in this plane, from the origin is proportional to

Options

A. `r^2`
B. `r`
C. `1//r`
D. `1//r^2`

Explanation

To solve the problem of determining how the electric field intensity at a distance \( r \) in the x-y plane is proportional to the given electric potential \( V = -kxy \), we will follow these steps:

### Step 1: Write down the expression for the electric potential
The electric potential \( V \) is given as:
\[ V = -kxy \]

### Step 2: Calculate the electric field components
The electric field \( \mathbf{E} \) is related to the electric potential \( V \) by the negative gradient:
\[ \mathbf{E} = -\nabla V \]
This means we need to find the partial derivatives of \( V \) with respect to \( x \) and \( y \).

### Step 3: Compute the partial derivative with respect to \( x \)
Calculating the partial derivative of \( V \) with respect to \( x \):
\[ \frac{\partial V}{\partial x} = -k \cdot y \]
Thus, the \( x \)-component of the electric field is:
\[ E_x = -\frac{\partial V}{\partial x} = ky \]

### Step 4: Compute the partial derivative with respect to \( y \)
Now, calculating the partial derivative of \( V \) with respect to \( y \):
\[ \frac{\partial V}{\partial y} = -k \cdot x \]
Thus, the \( y \)-component of the electric field is:
\[ E_y = -\frac{\partial V}{\partial y} = kx \]

### Step 5: Write the electric field vector
Now we can express the electric field vector:
\[ \mathbf{E} = E_x \hat{i} + E_y \hat{j} = ky \hat{i} + kx \hat{j} \]

### Step 6: Calculate the magnitude of the electric field
The magnitude of the electric field \( E \) can be calculated using the Pythagorean theorem:
\[ E = \sqrt{E_x^2 + E_y^2} = \sqrt{(ky)^2 + (kx)^2} = k\sqrt{y^2 + x^2} \]

### Step 7: Relate \( x \) and \( y \) to \( r \)
Since we are interested in the electric field at a distance \( r \) from the origin, we can use the relation:
\[ r^2 = x^2 + y^2 \implies \sqrt{x^2 + y^2} = r \]
Thus, we can express the magnitude of the electric field as:
\[ E = kr \]

### Step 8: Conclusion
The electric field intensity at a distance \( r \) from the origin is proportional to:
\[ E \propto r \]