Calculating the Net Force on a Charge at the Corner of an Equilateral Triangle

In this article, we will go through the process of calculating the net force on a charge placed at one corner of an equilateral triangle with identical charges at the other two corners. This problem involves the application of Coulomb's law and vector addition principles.

Given Data

For this problem, we have three identical charges, each of magnitude Q 6.39 × 10-4 C, positioned at the vertices of an equilateral triangle with side length a 3.35 m. The goal is to determine the magnitude of the net force on the charge at one corner of the triangle, which we will refer to as Q3.

Step 1: Calculate the Force Between Two Charges

The force between two point charges is given by Coulomb's law:

F k frac{q1q2}{r2}

where:

k 8.99 × 109 N m2/C2 is the Coulomb constant, r is the distance between the charges, in this case, r a 3.35 m.

Step 2: Calculate the Force Between Q3 and Q1 and Q2

The force between charges Q3 and Q1 (or Q2) is:

F31 k frac{Q^2}{a^2}

Substituting the given values:

F31 8.99 × 109 frac{6.39 × 10-42}{3.352}

Calculating F31:

F31 8.99 × 109 frac{4.079 × 10-7}{11.2225}

F31 ≈ 327.4 N

Since the charge Q2 is identical to Q1, the force between Q3 and Q2 is also:

F32 ≈ 327.4 N

Step 3: Determine the Direction of the Forces

The forces F31 and F32 act along the lines connecting the charges. Specifically:

F31 points away from Q1 towards Q3. F32 points away from Q2 towards Q3.

Step 4: Calculate the Net Force on Q3

The angle between the forces F31 and F32 is 60°, which is the internal angle of an equilateral triangle. To find the net force, we use vector addition principles:

Fnet sqrt{F312 F322 - 2F31F32cos120°}

Substituting the values:

Fnet sqrt{327.42 327.42 - 2 × 327.4 × 327.4 times -0.5}

Calculating:

Fnet sqrt{2 × 327.42 327.42}

Fnet sqrt{3 × 327.42}

Fnet ≈ 327.4 N

Final Answer

The magnitude of the net force on charge Q3 is approximately 327.4 N.

Alternative Method

Another approach to solving this problem is to consider the forces at the central charge due to the other two charges. In an equilateral triangle, the forces acting on one charge at the corner have magnitudes equal to F, with one force directed along AC and the other along BC. The resultant force is directed along MC, where M is the midpoint of BC. Each force has a component along MC, given by Fcos(30°). Therefore, the resultant force is 2Fcos(30°) or 3F.

The numerical values can be calculated as:

Coulomb constant, 1/4πε0 9 × 109 N m2/C2 Charge magnitude, Q 6.39 × 10-4 C Distance between charges, a 3.35 m

Using the forces to find the net force directly:

F 9 × 109 frac{(6.39 × 10-4 2}{3.352} 327.4 N

Resultant force:

2Fcos(30°) 2 × 327.4 × frac{sqrt{3}}{2} ≈ 327.4 sqrt{3} ≈ 567.3 N

The resultant force simplifies to approximately 327.4 N in magnitude.