Two charges of $-4\ \mu\text{C}$ and $+4\ \mu\text{C}$ are placed at the points $A(1, 0, 4)\ \text{m}$ and $B(2, -1, 5)\ \text{m}$ located in an electric field $\vec E = 0.20\,\hat i\ \text{V cm}^{-1}$. The magnitude of the torque acting on the dipole is $8\sqrt\alpha\times10^{-5}\ \text{N m}$, where $\alpha =$ ______.
Numerical value type. Enter your answer.
Answer: 2
The dipole moment points from the negative to the positive charge:
$$\vec p = q(\vec r_B - \vec r_A) = 4\times10^{-6}(\hat i - \hat j + \hat k)\ \text{C m}$$
$\vec E = 0.20\ \text{V cm}^{-1} = 20\,\hat i\ \text{V m}^{-1}$.
$$\vec\tau = \vec p\times\vec E = 8\times10^{-5}\,(\hat i - \hat j + \hat k)\times\hat i = 8\times10^{-5}(\hat j + \hat k)$$
$$|\vec\tau| = 8\sqrt2\times10^{-5}\ \text{N m} \;\Rightarrow\; \alpha = 2$$
Solution by Sreeraj P, M.Sc Physics