In any region, if electric field is define as `bar(E ) = ( hat(i) + 2 hat(j)+ hat(k))V//m` , then the potential differnece between two points `A ( 0,0,0)` and `B ( 2,3,4)` in that region is

Options

A. 6V
B. 12 V
C. 8 V
D. 9V

Answer

To find the potential difference between two points A(0, 0, 0) and B(2, 3, 4) in the given electric field \(\bar{E} = \hat{i} + 2\hat{j} + \hat{k}\) V/m, we can follow these steps:

### Step 1: Understand the relationship between electric field and potential difference
The potential difference (\(\Delta V\)) between two points in an electric field is given by the negative line integral of the electric field along the path from one point to the other:
\[ \Delta V = -\int_A^B \bar{E} \cdot d\bar{r} \]

### Step 2: Define the position vectors
The position vector of point A is:
\[ \bar{r}_A = 0\hat{i} + 0\hat{j} + 0\hat{k} = \vec{0} \]
The position vector of point B is:
\[ \bar{r}_B = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

### Step 3: Determine the differential displacement vector \(d\bar{r}\)
The displacement vector \(d\bar{r}\) from A to B can be expressed as:
\[ d\bar{r} = \bar{r}_B - \bar{r}_A = (2\hat{i} + 3\hat{j} + 4\hat{k}) - (0\hat{i} + 0\hat{j} + 0\hat{k}) = 2\hat{i} + 3\hat{j} + 4\hat{k} \]

### Step 4: Calculate the dot product \(\bar{E} \cdot d\bar{r}\)
Now, we calculate the dot product of the electric field \(\bar{E}\) and the displacement vector \(d\bar{r}\):
\[ \bar{E} = \hat{i} + 2\hat{j} + \hat{k} \]
\[ d\bar{r} = 2\hat{i} + 3\hat{j} + 4\hat{k} \]
Calculating the dot product:
\[ \bar{E} \cdot d\bar{r} = (1)(2) + (2)(3) + (1)(4) = 2 + 6 + 4 = 12 \]

### Step 5: Calculate the potential difference
Substituting the result of the dot product into the equation for potential difference:
\[ \Delta V = -\int_A^B \bar{E} \cdot d\bar{r} = -12 \, \text{V} \]

Thus, the potential difference between points A and B is:
\[ \Delta V = -12 \, \text{V} \]

### Summary of the Solution
The potential difference between points A(0, 0, 0) and B(2, 3, 4) in the electric field \(\bar{E} = \hat{i} + 2\hat{j} + \hat{k}\) is \(-12 \, \text{V}\).

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