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, we need to find the electric field intensity from the given electric potential \( V = -kxy \). The electric field \( \vec{E} \) is related to the electric potential \( V \) by the equation:

\[ \vec{E} = -\nabla V \]

Where \( \nabla V \) is the gradient of the potential \( V \). In Cartesian coordinates, this can be expressed as:

\[ \vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \right) \]

### Step 1: Calculate the partial derivatives of \( V \)

1. **Calculate \( \frac{\partial V}{\partial x} \)**:
\[ V = -kxy \implies \frac{\partial V}{\partial x} = -k \cdot y \]

2. **Calculate \( \frac{\partial V}{\partial y} \)**:
\[ V = -kxy \implies \frac{\partial V}{\partial y} = -k \cdot x \]

3. **Calculate \( \frac{\partial V}{\partial z} \)**:
Since \( V \) does not depend on \( z \):
\[ \frac{\partial V}{\partial z} = 0 \]

### Step 2: Substitute the derivatives into the electric field equation

Now we can substitute these derivatives into the electric field equation:

\[ \vec{E} = -\left( -k y \hat{i} - k x \hat{j} + 0 \hat{k} \right) \]

This simplifies to:

\[ \vec{E} = k y \hat{i} + k x \hat{j} \]

### Step 3: Find the magnitude of the electric field

The magnitude of the electric field \( |\vec{E}| \) can be calculated as follows:

\[ |\vec{E}| = \sqrt{(k y)^2 + (k x)^2} = k \sqrt{y^2 + x^2} \]

### Step 4: Express \( r \) in terms of \( x \) and \( y \)

We know that:

\[ r = \sqrt{x^2 + y^2} \]

Thus, we can rewrite the magnitude of the electric field as:

\[ |\vec{E}| = k r \]

### Conclusion

The electric field intensity at a distance \( r \) in the x-y plane from the origin is proportional to \( r \).

### Final Answer

The electric field intensity at a distance \( r \) is proportional to \( r \).

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