Two charges of $$-4 \mu$$C and $$+4 \mu$$C are placed at the points $$A(1, 0, 4)$$ m and $$B(2, -1, 5)$$ m located in an electric field $$\vec{E} = 0.20\hat{i}$$ V cm$$^{-1}$$. The magnitude of the torque acting on the dipole is $$8\sqrt{\alpha} \times 10^{-5}$$ N m, where $$\alpha$$ = _____.

Dipole moment

$$⃗\vec{p}=q\vec{d}$$

Position vector from −4μC to 4μC:

$$\vec{d}=(2-1)i+(-1-0)j+(5-4)k$$

$$=i−j+k$$

Charge

$$q=4\times10^{-6}C$$

So

Electric field:

$$E⃗=0.20iV/cm$$

Convert to SI:

$$0.20V/cm=20N/C$$

Torque:

$$τ=\vec{p}\times\vec{E}$$

Magnitude

$$\tau=pE\sin\theta$$

Only components perpendicular to i contribute:

$$p_{\perp}=4\times10^{-6}\sqrt{(-1)^2+1^2}$$

$$=4\sqrt{2}\times10^{-6}$$

Thus

$$\tau=(4\sqrt{2}\times10^{-6})(20)$$

$$=8\sqrt{2}\times10^{-5}$$