If the potentail in the region of space around the point `(-1m,2m,3m)` is given by `V = (10 x^(2) + 5 y^(2) - 3 z^(2))`, calculate the three components of electric field at this point.

Explanation

To calculate the electric field components at the point (-1m, 2m, 3m) given the potential \( V = 10x^2 + 5y^2 - 3z^2 \), we will follow these steps:

### Step 1: Understand the relationship between electric field and potential
The electric field \( \vec{E} \) is related to the electric potential \( V \) by the equation:
\[ \vec{E} = -\nabla V \]
This means that the electric field components in the x, y, and z directions can be found by taking the negative gradient of the potential.

### Step 2: Calculate the partial derivatives
We need to find the partial derivatives of \( V \) with respect to \( x \), \( y \), and \( z \).

1. **For the x-component:**
\[ E_x = -\frac{\partial V}{\partial x} = -\frac{\partial}{\partial x}(10x^2 + 5y^2 - 3z^2) \]
Since \( y \) and \( z \) are treated as constants, the derivative simplifies to:
\[ E_x = -20x \]

2. **For the y-component:**
\[ E_y = -\frac{\partial V}{\partial y} = -\frac{\partial}{\partial y}(10x^2 + 5y^2 - 3z^2) \]
Here, \( x \) and \( z \) are treated as constants, leading to:
\[ E_y = -10y \]

3. **For the z-component:**
\[ E_z = -\frac{\partial V}{\partial z} = -\frac{\partial}{\partial z}(10x^2 + 5y^2 - 3z^2) \]
With \( x \) and \( y \) as constants, we find:
\[ E_z = 6z \]

### Step 3: Evaluate the derivatives at the given point
Now we will substitute the coordinates \( x = -1 \), \( y = 2 \), and \( z = 3 \) into the expressions we derived.

1. **Calculate \( E_x \):**
\[ E_x = -20(-1) = 20 \, \text{N/C} \]

2. **Calculate \( E_y \):**
\[ E_y = -10(2) = -20 \, \text{N/C} \]

3. **Calculate \( E_z \):**
\[ E_z = 6(3) = 18 \, \text{N/C} \]

### Step 4: Write the final electric field vector
Combining these components, the electric field vector at the point (-1m, 2m, 3m) is:
\[ \vec{E} = 20 \hat{i} - 20 \hat{j} + 18 \hat{k} \, \text{N/C} \]

### Final Answer:
The three components of the electric field at the point (-1m, 2m, 3m) are:
\[ \vec{E} = 20 \hat{i} - 20 \hat{j} + 18 \hat{k} \, \text{N/C} \]

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