Determine the electric field strength vector if the potential of this field depends on X, coordinates as V = 10 a`x y `

Options

A. 10 a `(y hat(i) + x hat(j))`
B. `-10a [ 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 \( V = 10 A x y \), we can 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 gradient of the potential
The potential \( V \) is given as:
\[ V = 10 A x y \]
To find the electric field, we need to calculate the partial derivatives of \( V \) with respect to \( x \) and \( y \).

### Step 3: Differentiate \( V \) with respect to \( x \)
Calculating the partial derivative of \( V \) with respect to \( x \):
\[ \frac{\partial V}{\partial x} = 10 A y \]

### Step 4: Differentiate \( V \) with respect to \( y \)
Now, calculating the partial derivative of \( V \) with respect to \( y \):
\[ \frac{\partial V}{\partial y} = 10 A x \]

### Step 5: Formulate the electric field vector
Using the results from the partial derivatives, we can write the electric field vector:
\[ \vec{E} = -\left( \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} \right) \]
Substituting the derivatives we calculated:
\[ \vec{E} = -\left( 10 A y \hat{i} + 10 A x \hat{j} \right) \]
This simplifies to:
\[ \vec{E} = -10 A y \hat{i} - 10 A x \hat{j} \]

### Step 6: Write the final form of the electric field vector
We can factor out \(-10 A\):
\[ \vec{E} = -10 A (y \hat{i} + x \hat{j}) \]

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

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