In a uniform electric field, the potential is ` 10V` at the origin of coordinates , and ` 8 V` at each of the points (1, 0, 0), (0, 1, 0) and (0, 0, 1). The potential at the point (1, 1,1 ) will be .

Options

A. 0
B. `4V`
C. `8V`
D. `10V`

Explanation

To find the potential at the point (1, 1, 1) in a uniform electric field, we can follow these steps:

### Step 1: Understand the given information
We know that:
- The potential \( V \) at the origin (0, 0, 0) is \( 10 \, V \).
- The potential \( V \) at the points (1, 0, 0), (0, 1, 0), and (0, 0, 1) is \( 8 \, V \).

### Step 2: Calculate the electric field components
Using the relationship between potential difference and electric field:
\[ \Delta V = -\mathbf{E} \cdot \Delta \mathbf{r} \]
For the point (1, 0, 0):
\[ \Delta V = V(1, 0, 0) - V(0, 0, 0) = 8 - 10 = -2 \, V \]
The displacement \( \Delta \mathbf{r} \) is \( (1, 0, 0) \), so:
\[ -2 = -E_x \cdot 1 \implies E_x = 2 \, \text{V/m} \]

For the point (0, 1, 0):
\[ \Delta V = V(0, 1, 0) - V(0, 0, 0) = 8 - 10 = -2 \, V \]
The displacement \( \Delta \mathbf{r} \) is \( (0, 1, 0) \), so:
\[ -2 = -E_y \cdot 1 \implies E_y = 2 \, \text{V/m} \]

For the point (0, 0, 1):
\[ \Delta V = V(0, 0, 1) - V(0, 0, 0) = 8 - 10 = -2 \, V \]
The displacement \( \Delta \mathbf{r} \) is \( (0, 0, 1) \), so:
\[ -2 = -E_z \cdot 1 \implies E_z = 2 \, \text{V/m} \]

### Step 3: Write the electric field vector
The electric field vector \( \mathbf{E} \) can be expressed as:
\[ \mathbf{E} = 2 \hat{i} + 2 \hat{j} + 2 \hat{k} \, \text{V/m} \]

### Step 4: Calculate the potential at (1, 1, 1)
Now, we need to find the potential \( V' \) at the point (1, 1, 1):
\[ \Delta \mathbf{r} = (1, 1, 1) - (0, 0, 0) = (1, 1, 1) \]
Using the potential difference formula:
\[ \Delta V = -\mathbf{E} \cdot \Delta \mathbf{r} \]
Calculating the dot product:
\[ \Delta V = -\left(2 \hat{i} + 2 \hat{j} + 2 \hat{k}\right) \cdot (1 \hat{i} + 1 \hat{j} + 1 \hat{k}) = -\left(2 \cdot 1 + 2 \cdot 1 + 2 \cdot 1\right) = -6 \, V \]
Thus:
\[ V' - V(0, 0, 0) = -6 \implies V' - 10 = -6 \implies V' = 10 - 6 = 4 \, V \]

### Final Answer
The potential at the point (1, 1, 1) is \( 4 \, V \).

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