To find the volume charge density at the point (-1, 0, 3) given the electric field \( \mathbf{E} \), we can use Gauss's law in its differential form, which states:
\[
\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}
\]
where \( \nabla \cdot \mathbf{E} \) is the divergence of the electric field, \( \rho \) is the volume charge density, and \( \epsilon_0 \) is the permittivity of free space.
### Step 1: Write down the electric field
The electric field is given as:
\[
\mathbf{E} = \frac{1}{\epsilon_0} \left[ (2y^2 + z) \hat{i} + 4xy \hat{j} + x \hat{k} \right] \, \text{V/m}
\]
### Step 2: Calculate the divergence of \( \mathbf{E} \)
The divergence operator in three dimensions is given by:
\[
\nabla \cdot \mathbf{E} = \frac{\partial E_x}{\partial x} + \frac{\partial E_y}{\partial y} + \frac{\partial E_z}{\partial z}
\]
where \( E_x = \frac{1}{\epsilon_0}(2y^2 + z) \), \( E_y = \frac{1}{\epsilon_0}(4xy) \), and \( E_z = \frac{1}{\epsilon_0}(x) \).
### Step 3: Compute each partial derivative
1. **For \( E_x \)**:
\[
\frac{\partial E_x}{\partial x} = \frac{\partial}{\partial x} \left( \frac{1}{\epsilon_0}(2y^2 + z) \right) = 0
\]
(since \( 2y^2 + z \) does not depend on \( x \)).
2. **For \( E_y \)**:
\[
\frac{\partial E_y}{\partial y} = \frac{\partial}{\partial y} \left( \frac{1}{\epsilon_0}(4xy) \right) = \frac{4x}{\epsilon_0}
\]
3. **For \( E_z \)**:
\[
\frac{\partial E_z}{\partial z} = \frac{\partial}{\partial z} \left( \frac{1}{\epsilon_0}(x) \right) = 0
\]
(since \( x \) does not depend on \( z \)).
### Step 4: Combine the results
Now, substituting these results into the divergence equation:
\[
\nabla \cdot \mathbf{E} = 0 + \frac{4x}{\epsilon_0} + 0 = \frac{4x}{\epsilon_0}
\]
### Step 5: Relate divergence to charge density
From Gauss's law, we have:
\[
\nabla \cdot \mathbf{E} = \frac{\rho}{\epsilon_0}
\]
Thus,
\[
\frac{4x}{\epsilon_0} = \frac{\rho}{\epsilon_0}
\]
This simplifies to:
\[
\rho = 4x
\]
### Step 6: Evaluate \( \rho \) at the point (-1, 0, 3)
Now, substitute \( x = -1 \):
\[
\rho = 4(-1) = -4 \, \text{C/m}^3
\]
### Final Answer
The volume charge density at the point (-1, 0, 3) is:
\[
\rho = -4 \, \text{C/m}^3
\]
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