Potential (V) at a point in space is given by `v=x^(2)+y^(2)+z^(2)`. Gravitational field at a point (x,y,z) is

Options

A. `-2x hat(i)-2y hat(j)-2z hat(k)`
B. `2x hat(i)+2y hat(j)+2z hat(k)`
C. `x hat(i)+y hat(j)+ z hat(k)`
D. `-x hat(i)-y hat(j)-z hat(k)`

Answer

To find the gravitational field at a point (x, y, z) given the potential \( V = x^2 + y^2 + z^2 \), we will follow these steps:

### Step 1: Understand the relationship between potential and gravitational field
The gravitational field \( \mathbf{E} \) is related to the gravitational potential \( V \) by the equation:
\[ \mathbf{E} = -\nabla V \]
where \( \nabla \) (nabla) is the gradient operator.

### Step 2: Calculate the gradient of the potential
The gradient operator in three dimensions is defined as:
\[ \nabla V = \frac{\partial V}{\partial x} \hat{i} + \frac{\partial V}{\partial y} \hat{j} + \frac{\partial V}{\partial z} \hat{k} \]

### Step 3: Differentiate the potential with respect to x, y, and z
Now, we will compute the partial derivatives of \( V \):
1. **With respect to x**:
\[ \frac{\partial V}{\partial x} = \frac{\partial}{\partial x}(x^2 + y^2 + z^2) = 2x \]
2. **With respect to y**:
\[ \frac{\partial V}{\partial y} = \frac{\partial}{\partial y}(x^2 + y^2 + z^2) = 2y \]
3. **With respect to z**:
\[ \frac{\partial V}{\partial z} = \frac{\partial}{\partial z}(x^2 + y^2 + z^2) = 2z \]

### Step 4: Write the gradient vector
Now substituting these derivatives into the gradient expression:
\[ \nabla V = 2x \hat{i} + 2y \hat{j} + 2z \hat{k} \]

### Step 5: Calculate the gravitational field
Now, substituting the gradient into the equation for the gravitational field:
\[ \mathbf{E} = -\nabla V = - (2x \hat{i} + 2y \hat{j} + 2z \hat{k}) \]
This simplifies to:
\[ \mathbf{E} = -2x \hat{i} - 2y \hat{j} - 2z \hat{k} \]

### Final Result
Thus, the gravitational field at the point \( (x, y, z) \) is:
\[ \mathbf{E} = -2x \hat{i} - 2y \hat{j} - 2z \hat{k} \]