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Coulomb's Law

CSCA Coulomb's Law study guide organized around the publicly available CSCA syllabus. Practice Physics questions on aicsca.com.

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Topic: Electromagnetism > Electrostatics > Coulomb's Law

1. Core Definition

**Coulomb's Law** is the foundation of electrostatics. It quantitatively describes the interaction force (Electrostatic Force or Coulomb Force) between two **stationary point charges** in a **vacuum**.

* **Conditions**:

1. **Vacuum** (Approximately valid in air).

2. **Point Charge**: The size/shape of the object is negligible compared to the distance ($r \gg$ size).

3. **Stationary**.

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2. Core Formula

$$F = k \frac{|q_1 q_2|}{r^2}$$

| Symbol | Meaning | Unit | Note |

| :---: | :--- | :---: | :--- |

| $F$ | Magnitude of Force | $N$ | Obeys Newton's 3rd Law (Action-Reaction) |

| $q$ | Charge Magnitude | $C$ | **Use Absolute Values for calculation** |

| $r$ | Distance | $m$ | Distance between **centers** of charges |

| $k$ | Coulomb's Constant | - | $k \approx 9.0 \times 10^9 \, N\cdot m^2/C^2$ |

**CSCA Exam Tips**:

1. **Unit Traps**: Problems often use micro-Coulombs ($\|mu C$) or centimeters ($cm$). You **must** convert to $C$ ($1\|mu C = 10^{-6}C$) and $m$ before calculating.

2. **Absolute Value**: The formula calculates magnitude only. Determine direction separately based on signs.

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3. Determining Direction

Follow the rule: **"Like charges repel, Unlike charges attract"**. The force acts along the line connecting the charges.

* **Repulsion**: $++$ or $--$

* **Attraction**: $+-$ or $-+$

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4. Key Method: Superposition of Forces

When a charge is acted upon by multiple charges, use the **Vector Superposition Principle** (Parallelogram Rule).

**Standard Steps:**

1. **Free Body Diagram**: Identify the object and draw force vectors.

2. **Calculate Magnitudes**: Use $F = k|q_1 q_2|/r^2$ for each pair.

3. **Determine Directions**: Mark the direction of each force vector.

4. **Vector Addition**:

* **Collinear**: Add or subtract algebraically.

* **Non-Collinear**: Use geometry (Pythagorean theorem, trigonometry) or component analysis.

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5. Typical Examples

**Example 1: Basic Calculation (Collinear)**

**Problem**: Three point charges are fixed on the x-axis. $q_A = +2.0 \times 10^{-9}C$ at $x=0$, $q_B = -3.0 \times 10^{-9}C$ at $x=0.1m$, $q_C = +4.0 \times 10^{-9}C$ at $x=0.2m$. Find the net force on $q_B$.

**Solution**:

1. **Analyze $q_B$**:

* $A$ on $B$ (Attraction): $F_{AB}$ points to $A$ (Left).

* $C$ on $B$ (Attraction): $F_{CB}$ points to $C$ (Right).

2. **Calculate Magnitudes**:

$$F_{AB} = 5.4 \times 10^{-6} N$$

$$F_{CB} = 1.08 \times 10^{-5} N$$

3. **Net Force**:

Taking right as positive.

$$F_{net} = F_{CB} - F_{AB} = 5.4 \times 10^{-6} N$$

Direction: **Right**.

**Example 2: Vector Property (Newton's 3rd Law)**

**Problem**: Charge $A$ is 3 times larger than $B$ ($q_A = 3q_B$). If the force exerted by $A$ on $B$ is $F_1$, and by $B$ on $A$ is $F_2$, compare $F_1$ and $F_2$.

**Solution**: According to Newton's 3rd Law, **$F_1$ is always equal to $F_2$**, regardless of the charge magnitudes.

**Example 3: Non-Collinear (Right Triangle)**

**Problem**: In a vacuum, $q_1=+Q$ is at $(0, a)$ and $q_2=+Q$ is at $(a, 0)$. Find the net force on a charge $-q$ placed at the origin $O(0,0)$.

**Solution**:

1. $q_1$ pulls $-q$ Up ($+y$) with $F_1 = kQq/a^2$.

2. $q_2$ pulls $-q$ Right ($+x$) with $F_2 = kQq/a^2$.

3. $F_1 \perp F_2$, so $F_{net} = \sqrt{F_1^2 + F_2^2} = \sqrt{2} k \frac{Qq}{a^2}$.

Direction: $45^\circ$ (towards the first quadrant bisector).