Find `V_(ab)` in an electric field `vec(E)=(2hat(i)+3hat(j)+4hat(k))N//C` where `vec(r)_(a)=(hat(i)-2hat(j)+hat(k))mandvec(r_(b))=(2hat(i)+hat(j)-2hat(k))m`

Options

A. 2 volt
B. `-1` volt
C. 1 volt
D. 3 volt

Answer

To find the potential difference \( V_{ab} \) in the given electric field, we can follow these steps:

### Step 1: Understand the relationship between electric field and potential difference
The potential difference \( V_{ab} \) between points \( a \) and \( b \) in an electric field \( \vec{E} \) is given by the equation:
\[ V_{ab} = V_a - V_b = -\int_{b}^{a} \vec{E} \cdot d\vec{r} \]
where \( d\vec{r} \) is the differential displacement vector along the path from point \( b \) to point \( a \).

### Step 2: Define the electric field and position vectors
The electric field is given as:
\[ \vec{E} = 2\hat{i} + 3\hat{j} + 4\hat{k} \, \text{N/C} \]
The position vectors for points \( a \) and \( b \) are:
\[ \vec{r}_a = \hat{i} - 2\hat{j} + \hat{k} \, \text{m} \]
\[ \vec{r}_b = 2\hat{i} + \hat{j} - 2\hat{k} \, \text{m} \]

### Step 3: Determine the limits of integration
We need to calculate the displacement vector \( d\vec{r} \) from point \( b \) to point \( a \):
\[ d\vec{r} = \vec{r}_a - \vec{r}_b = (\hat{i} - 2\hat{j} + \hat{k}) - (2\hat{i} + \hat{j} - 2\hat{k}) = -\hat{i} - 3\hat{j} + 3\hat{k} \]

### Step 4: Set up the integral
Now we can express the integral:
\[ V_{ab} = -\int_{b}^{a} \vec{E} \cdot d\vec{r} \]
Substituting \( \vec{E} \) and \( d\vec{r} \):
\[ V_{ab} = -\int_{b}^{a} (2\hat{i} + 3\hat{j} + 4\hat{k}) \cdot (-\hat{i} - 3\hat{j} + 3\hat{k}) \, dr \]

### Step 5: Calculate the dot product
Calculating the dot product:
\[ \vec{E} \cdot d\vec{r} = 2(-1) + 3(-3) + 4(3) = -2 - 9 + 12 = 1 \]

### Step 6: Integrate
Since the dot product is constant, we can integrate directly:
\[ V_{ab} = -\int_{b}^{a} 1 \, dr = -\left[r\right]_{b}^{a} = -\left(a - b\right) \]

### Step 7: Evaluate the limits
The limits of integration are the coordinates of points \( a \) and \( b \):
\[ V_{ab} = -\left(1 - 2\right) = -(-1) = 1 \, \text{volt} \]

### Final Answer
Thus, the potential difference \( V_{ab} \) is:
\[ V_{ab} = 1 \, \text{volt} \]

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