Two charges of `-4 muC and +4muC` are placed at the points `A (1,0,4) and B (2,-1,5)` location in an electric field `vec(E) = 0.20 hat(i) V//cm`. Calculate the torque acting on the dipole.

Options

A. `2.31 xx 10^(-4) N//m`
B. `1.13 xx 10^(-4) C//m`
C. `7.11 xx 10^(-4) C//m`
D. `7.04 xx 10^(-4) C//m`

Answer

To calculate the torque acting on the dipole formed by two charges of \(-4 \mu C\) and \(+4 \mu C\) placed at points \(A(1,0,4)\) and \(B(2,-1,5)\) in an electric field \(\vec{E} = 0.20 \hat{i} \, \text{V/cm}\), we can follow these steps:

### Step 1: Calculate the distance vector \( \vec{AB} \)
The distance vector \( \vec{AB} \) can be calculated as:
\[ \vec{AB} = \vec{B} - \vec{A} = (2, -1, 5) - (1, 0, 4) = (2-1, -1-0, 5-4) = (1, -1, 1) \]

### Step 2: Calculate the dipole moment \( \vec{P} \)
The dipole moment \( \vec{P} \) is given by:
\[ \vec{P} = q \cdot \vec{d} \]
where \( q = 4 \mu C = 4 \times 10^{-6} \, C \) and \( \vec{d} = \vec{AB} \). Therefore:
\[ \vec{P} = 4 \times 10^{-6} \, C \cdot (1, -1, 1) = (4 \times 10^{-6}, -4 \times 10^{-6}, 4 \times 10^{-6}) \, C \cdot m \]

### Step 3: Convert the electric field to SI units
The electric field is given as:
\[ \vec{E} = 0.20 \hat{i} \, \text{V/cm} = 0.20 \times 100 \, \hat{i} \, \text{V/m} = 20 \, \hat{i} \, \text{V/m} \]

### Step 4: Calculate the torque \( \vec{\tau} \)
The torque \( \vec{\tau} \) acting on the dipole in an electric field is given by:
\[ \vec{\tau} = \vec{P} \times \vec{E} \]
Substituting the values we have:
\[ \vec{\tau} = (4 \times 10^{-6}, -4 \times 10^{-6}, 4 \times 10^{-6}) \times (20, 0, 0) \]

### Step 5: Compute the cross product
Using the determinant method to calculate the cross product:
\[ \vec{\tau} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ 4 \times 10^{-6} & -4 \times 10^{-6} & 4 \times 10^{-6} \\ 20 & 0 & 0 \end{vmatrix} \]
Calculating this determinant:
\[ \vec{\tau} = \hat{i} \left((-4 \times 10^{-6}) \cdot 0 - (4 \times 10^{-6}) \cdot 0\right) - \hat{j} \left((4 \times 10^{-6}) \cdot 0 - (4 \times 10^{-6}) \cdot 20\right) + \hat{k} \left((4 \times 10^{-6}) \cdot 0 - (-4 \times 10^{-6}) \cdot 20\right) \]
\[ = 0 \hat{i} + 80 \times 10^{-6} \hat{j} + 80 \times 10^{-6} \hat{k} \]
Thus:
\[ \vec{\tau} = (0, 80 \times 10^{-6}, 80 \times 10^{-6}) \, N \cdot m \]

### Step 6: Calculate the magnitude of torque
The magnitude of torque is given by:
\[ |\vec{\tau}| = \sqrt{(0)^2 + (80 \times 10^{-6})^2 + (80 \times 10^{-6})^2} = \sqrt{2 \times (80 \times 10^{-6})^2} = 80 \times 10^{-6} \sqrt{2} \approx 1.13 \times 10^{-4} \, N \cdot m \]

### Final Answer
The torque acting on the dipole is approximately:
\[ \vec{\tau} \approx 1.13 \times 10^{-4} \, N \cdot m \]