Determine the electric field strength vector if the potential of this field depends on X, coordinates as V = 10 axy

Options

A. -10 a `(y hat(i) + x hat(j))`
B. `-10 [ y hat(i) + x hat(j)] `
C. ` - a [ y hat(i) + x hat(j)]`
D. `- 10 a [x hat(i) + y hat(k)] `

Answer

To determine the electric field strength vector from the given electric potential \( V = 10Axy \), we follow these steps:

### Step 1: Understand the relationship between electric field and electric potential
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.

### Step 2: Calculate the components of the electric field
The electric field has two components in the Cartesian coordinates (x and y). We can express it as:
\[ \vec{E} = E_x \hat{i} + E_y \hat{j} \]

### Step 3: Find the x-component of the electric field
To find \( E_x \), we need to compute:
\[ E_x = -\frac{\partial V}{\partial x} \]
Given \( V = 10Axy \), we differentiate with respect to \( x \):
\[ \frac{\partial V}{\partial x} = 10Ay \]
Thus,
\[ E_x = -10Ay \]

### Step 4: Find the y-component of the electric field
Next, we compute \( E_y \):
\[ E_y = -\frac{\partial V}{\partial y} \]
Differentiating \( V = 10Axy \) with respect to \( y \):
\[ \frac{\partial V}{\partial y} = 10Ax \]
So,
\[ E_y = -10Ax \]

### Step 5: Write the electric field vector
Now we can write the electric field vector:
\[ \vec{E} = E_x \hat{i} + E_y \hat{j} = (-10Ay) \hat{i} + (-10Ax) \hat{j} \]
This can be simplified to:
\[ \vec{E} = -10A(y \hat{i} + x \hat{j}) \]

### Final Answer
Thus, the electric field strength vector is:
\[ \vec{E} = -10A(y \hat{i} + x \hat{j}) \]

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