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Chapter 22: Problem 50

Three equal charges \(q\) form an equilateral triangle of side \(a\). Find the potential at the center of the triangle.

Short Answer

Expert verified
The electric potential at the center of the equilateral triangle formed by the three equal charges is given by \( V = 3k \cdot \dfrac{2q}{\sqrt{3}a}\).

Step by step solution

01

Identifying Values and Formulating the Equation

First identify the known values from the problem. We have three equal charges \(q\) and they form an equilateral triangle with each side of length \(a\). The electric potential formula is tailored to find the potential at the center of the triangle. As the triangle is equilateral, the distance from each charge to the center is equal to the height of the triangle, which can be calculated using Pythagoras' theorem.
02

Calculating the height of the triangle

The height of the triangle can be calculated using the Pythagorean theorem in relation to half the base of the triangle and the triangle side which is \(a\). Since it's an equilateral triangle, the half base is \(\dfrac{a}{2}\). Applying Pythagorean theorem: \(h = \sqrt{a^{2} - \left(\dfrac{a}{2}\right)^{2}} = \dfrac{\sqrt{3}}{2}a\). So, the distance from each charge to the center of the triangle is \(\dfrac{\sqrt{3}}{2}a\).
03

Calculating the Electric Potential

To calculate the electric potential at the center of the triangle caused by each of the three charges, substitute the values into the electric potential formula: \( V_{q} = k \cdot \dfrac{q}{h} = k \cdot \dfrac{q}{\dfrac{\sqrt{3}}{2}a}\). Since there are three identical charges, the total electric potential \( V \) at the center of the triangle is \( V = 3V_{q} = 3k \cdot \dfrac{2q}{\sqrt{3}a}\). Simplifying to get the final equation for the electric potential at the center of the triangle by the three equal point charges.

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Most popular questions from this chapter

A thin spherical shell has radius \(R\) and total charge \(Q\) distributed uniformly over its surface. Find the potential at its center.

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Must the potential be zero at any point where the electric field is zero? Explain.
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