The electirc potential at a point `(x, y, z)` is given by
`V = -x^(2)y - xz^(3) + 4`
The electric field `vecE` at that point is

Options

A. `vecE = hati (2xy +z^(3)) + hatj x^(2) + hatk 3xz^(2)`
B. `vecE = hati 2xy + hatj (x^(2) + y^(2)) + hatk (3xz - y^(2))`
C. `vecE = hati z^(3) + hatj xyz + hatk z^(2)`
D. `vecE = hati (2xy - z^(3)) + hatj xy^(2) + hatk 3z^(2) x`

Explanation

To find the electric field \(\vec{E}\) at the point \((x, y, z)\) given the electric potential \(V\), we can use the relation:

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

where \(\nabla V\) is the gradient of the potential \(V\). The gradient in three dimensions is given by:

\[ \nabla V = \left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]

Given the potential:

\[ V = -x^2 y - x z^3 + 4 \]

we will compute the partial derivatives with respect to \(x\), \(y\), and \(z\).

### Step 1: Calculate \(\frac{\partial V}{\partial x}\)

Differentiating \(V\) with respect to \(x\):

\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(-x^2 y - x z^3 + 4) \]

Using the product rule:

\[ \frac{\partial V}{\partial x} = -2xy - z^3 \]

### Step 2: Calculate \(\frac{\partial V}{\partial y}\)

Differentiating \(V\) with respect to \(y\):

\[ \frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(-x^2 y - x z^3 + 4) \]

Here, \( -x z^3 + 4\) is treated as a constant:

\[ \frac{\partial V}{\partial y} = -x^2 \]

### Step 3: Calculate \(\frac{\partial V}{\partial z}\)

Differentiating \(V\) with respect to \(z\):

\[ \frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(-x^2 y - x z^3 + 4) \]

The first term is constant with respect to \(z\):

\[ \frac{\partial V}{\partial z} = -3xz^2 \]

### Step 4: Combine the results to find \(\vec{E}\)

Now we can substitute the partial derivatives back into the expression for \(\vec{E}\):

\[ \vec{E} = -\nabla V = -\left( \frac{\partial V}{\partial x}, \frac{\partial V}{\partial y}, \frac{\partial V}{\partial z} \right) \]

Substituting the values we found:

\[ \vec{E} = -\left( -2xy - z^3, -x^2, -3xz^2 \right) \]

This simplifies to:

\[ \vec{E} = (2xy + z^3) \hat{i} + x^2 \hat{j} + 3xz^2 \hat{k} \]

### Final Answer

Thus, the electric field \(\vec{E}\) at the point \((x, y, z)\) is:

\[ \vec{E} = (2xy + z^3) \hat{i} + x^2 \hat{j} + 3xz^2 \hat{k} \]

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