Given: electric potential, `phi = x^(2) + y^(2) +z^(2)`. The modulus of electric field at `(x, y, z)` is

Options

A. `2(x+ y+ z)`
B. `2sqrt(x^(2) + y^(2) +z^(2))`
C. `2x + y + z`
D. `xyz`

Explanation

To find the modulus of the electric field given the electric potential \( \phi = x^2 + y^2 + z^2 \), we can follow these steps:

### Step 1: Understand the relationship between electric potential and electric field
The electric field \( \mathbf{E} \) can be derived from the electric potential \( \phi \) using the formula:
\[ \mathbf{E} = -\nabla \phi \]
where \( \nabla \phi \) is the gradient of the potential.

### Step 2: Calculate the gradient of \( \phi \)
The gradient in three dimensions is given by:
\[ \nabla \phi = \left( \frac{\partial \phi}{\partial x}, \frac{\partial \phi}{\partial y}, \frac{\partial \phi}{\partial z} \right) \]
Now, we will compute each component of the gradient.

### Step 3: Compute \( \frac{\partial \phi}{\partial x} \)
Differentiating \( \phi \) with respect to \( x \):
\[ \frac{\partial \phi}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2 + z^2) = 2x \]

### Step 4: Compute \( \frac{\partial \phi}{\partial y} \)
Differentiating \( \phi \) with respect to \( y \):
\[ \frac{\partial \phi}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2 + z^2) = 2y \]

### Step 5: Compute \( \frac{\partial \phi}{\partial z} \)
Differentiating \( \phi \) with respect to \( z \):
\[ \frac{\partial \phi}{\partial z} = \frac{\partial}{\partial z}(x^2 + y^2 + z^2) = 2z \]

### Step 6: Write the electric field components
Now, substituting these derivatives into the expression for the electric field:
\[ \mathbf{E} = -\nabla \phi = \left(-\frac{\partial \phi}{\partial x}, -\frac{\partial \phi}{\partial y}, -\frac{\partial \phi}{\partial z}\right) = (-2x, -2y, -2z) \]

### Step 7: Calculate the modulus of the electric field
The modulus (magnitude) of the electric field \( |\mathbf{E}| \) is given by:
\[ |\mathbf{E}| = \sqrt{E_x^2 + E_y^2 + E_z^2} \]
Substituting the components:
\[ |\mathbf{E}| = \sqrt{(-2x)^2 + (-2y)^2 + (-2z)^2} = \sqrt{4x^2 + 4y^2 + 4z^2} \]
This simplifies to:
\[ |\mathbf{E}| = \sqrt{4(x^2 + y^2 + z^2)} = 2\sqrt{x^2 + y^2 + z^2} \]

### Final Answer
Thus, the modulus of the electric field at the point \((x, y, z)\) is:
\[ |\mathbf{E}| = 2\sqrt{x^2 + y^2 + z^2} \]

---