Question

Three particles, each with positive chargeQ, form an equilateral triangle, with each side of length d.What is the magnitude of the electric field produced by the particles at the midpoint of any side?

Short answer

The magnitude of the electric field is.$\frac{4}{3}\frac{Q}{4\pi {\epsilon }_o{d}^2}$

Step-by-step solution

The given data

Three particles each with positive chargeQ, form an equilateral triangle of side length,d .

Understanding the concept of electric field

Using the concept of the electric field between two charges, the required value of the magnitude of the electric field can be given.

Formula:

The magnitude of the electric field of the system of charges, $E=\frac{Q}{4\pi {\epsilon }_o{r}^2}$ (i)

Calculation of the magnitude of the electric field

The two closest charges produce fields at the midpoint that cancel each other out. Thus, the only significant contribution is from the charge which is at a distance,$r=\sqrt{3}d/2$away from the mid-point. Substituting the given values in equation (i), we get the magnitude of the electric field as follows:

$\begin{array}{c}E=\frac{Q}{4\pi {\epsilon }_o{(\sqrt{3}d/2)}^2}\\ =\frac{4}{3}\frac{Q}{4\pi {\epsilon }_o{d}^2}\end{array}$

Hence, the value of the electric field is.$\frac{4}{3}\frac{Q}{4\pi {\epsilon }_o{d}^2}$