Question

Three points charges are placed at the corners of an equilateral triangle of side $L$ shown in the figure.

• A. The potential at the centroid of the triangle is zero
• B. The electric field at the centroid of the triangle is zero
• C. The dipole moment of the system is $\sqrt{2}qL$
• D. The dipole moment of the system is $\sqrt{3}qL$

Answer 1

Answer: (A), (D)

The potential at the centroid of the triangle is zero
The dipole moment of the system is $\sqrt{3}qL$

For equilateral triangle AOB, $AO=OB=BA=a$ and $AC=OC=BC=r$

The potential at centroid C is $V=k\frac{2q}{AC}+k\frac{-q}{OC}+k\frac{-q}{BC}=k\frac{2q}{r}-k\frac{q}{r}-k\frac{q}{r}=0$

The net electric field at centroid C is $E=k\frac{2q}{r^2}-k\frac{q}{r^2}\cos 30-k\frac{q}{r^2}\cos 30=k\left(0.2\right)\frac{2q}{r^2}\ne 0$

As the total charge is zero so for dipole moment the choice of origin is independent. We assume O as origin.

dipole moment , $p=-q\left(0\right)-q\left(L\hat{i}\right)+2q(\frac{L}{2}\hat{i}+\frac{\sqrt{3}L}{2}\hat{j})$

or $p=\sqrt{3}qL\hat{j}$