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Concept

Forces Between Multiple Charges

States the principle of superposition for electric forces.

The "Crowd" of Charges

Imagine being in a crowded room where several friends are pulling your arms in different directions. Your final movement depends on the combined pull of everyone.

Coulomb’s law is perfect for calculating the force between exactly two stationary charges. But what happens when a charge is surrounded by a whole system of other charges?

To find the net force in these real-world scenarios, we use a simple but powerful rule called the principle of superposition.

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Vector Setup

Vector Addition Setup

Breaks down how to construct the vector sum practically.

The Principle of Superposition

Imagine a tug-of-war where multiple people pull on a single object from different directions. To find out where the object goes, you can't just add their strengths as plain numbers; you must account for their pull directions.

In electrostatics, the principle of superposition handles this exact scenario. It states that the total force on a given charge is the vector sum of the individual forces exerted by all other surrounding charges.

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Diagram

Superposition Vectors

Visual of multiple force vectors acting on a single charge.

Clean scientific diagram of a system of three stationary point charges q1, q2, and q3. Show position vectors r1, r2, and r3 pointing from the origin O to each charge. Draw individual force vectors F12 and F13 acting on q1, and clearly use dotted lines to form a parallelogram to show the resultant vector sum force F1.

Click to zoom

The total force on a charge is the vector sum of individual pair-wise forces, resolved using geometric addition.

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Superposition Equation

Vector sum formula for multiple charges.

Superposition Principle

F_{1} = F_{12} + F_{13} + \dots + F_{1 n} = \frac{q _{1}}{4 π ε _{0}} i = 2 \sum n \frac{q _{i}}{r _{1 i}^{2}} \overset{r}{^}_{1 i}

The net force on one charge is found by adding all individual forces as vectors.

Symbol Meaning

F_{1}

— net vector force on charge

q_{1}

F_{12}, F_{13}, \dots

— forces on

q_{1}

due to other charges

q_{1}, q_{i}

— point charges in the system

r_{1 i}

— distance between

q_{1}

and

q_{i}

\overset{r}{^}_{1 i}

— unit vector showing the direction of force

Key Idea

This is a vector sum. So, forces must be added with their directions, not just by adding their magnitudes. The direction is included through $\overset{r}{^}_{1 i}$ .

Use vector addition, such as the parallelogram law or component method, when forces act in different directions.

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Worked Example

Equilateral Triangle

Calculates force on a central charge in a triangle.

Problem: Force at Centroid of an Equilateral Triangle

Consider three charges $q_1, q_2, q_3$ each equal to $q$ at the vertices of an equilateral triangle of side $l$ . What is the force on a charge $Q$ (with the same sign as $q$ ) placed at the centroid of the triangle, as shown in Fig. 1.6?

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Worksheet

Charges on a Triangle

Practice finding net force using symmetry.

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Consider three charges $q, q,$ and $-q$ placed at the vertices A, B, and C of an equilateral triangle of side $l$ . Let the magnitude of the electrostatic force between any pair of these charges be denoted as $F = \frac{1}{4\pi\varepsilon_0} \frac{q^2}{l^2}$ .

To find the net force $\mathbf{F}_1$ on the charge $q$ at vertex A, we sum the repulsive force from B and the attractive force from C. Since the interior angle of the triangle is $60^\circ$ , the angle between these two force vectors is $120^\circ$ .

Using the parallelogram law, the magnitude is evaluated as $\sqrt{F^2 + F^2 + 2F^2 \cos(120^\circ)} =$ . For the charge $-q$ at vertex C, the forces from A and B are both attractive, with an angle of $60^\circ$ between them.

The magnitude of the resultant force $\mathbf{F}_3$ is calculated as $\sqrt{F^2 + F^2 + 2F^2 \cos(60^\circ)} =$ .

Following Newton's third law, the vector sum of the forces on all three charges must equal .