Electric Field Due to Point Charges

Introduction

Key Concepts

1. Electric Charge and Coulomb's Law

Electric charge is a basic property of matter responsible for electric phenomena. Charges come in two types: positive and negative. Like charges repel each other, while opposite charges attract. The interaction between two point charges is quantitatively described by Coulomb's Law.

Coulomb's Law states that the electric force ($F$) between two point charges is directly proportional to the product of the charges ($q_1$ and $q_2$) and inversely proportional to the square of the distance ($r$) between them: $$ F = k_e \frac{q_1 q_2}{r^2} $$ where $k_e$ is Coulomb's constant ($8.988 \times 10^9 \, \text{N m}^2/\text{C}^2$).

2. Definition of Electric Field

The electric field ($\vec{E}$) is a vector field that represents the force experienced by a unit positive charge placed at a point in space. It quantifies the presence and influence of electric charges on their surroundings.

Mathematically, the electric field is defined as: $$ \vec{E} = \frac{\vec{F}}{q} $$ where $\vec{F}$ is the force exerted on a test charge $q$.

3. Electric Field Due to a Point Charge

For a single point charge ($q$), the electric field it creates at a distance ($r$) from the charge is radially outward (if the charge is positive) or inward (if the charge is negative). The magnitude of this electric field is given by: $$ E = k_e \frac{|q|}{r^2} $$ The direction is along the line connecting the charge and the point of interest.

4. Superposition Principle

When multiple point charges are present, the resultant electric field at any point is the vector sum of the electric fields produced by each individual charge. This principle allows us to analyze complex charge distributions by breaking them down into simpler components.

If there are charges $q_1, q_2, \ldots, q_n$, the total electric field ($\vec{E}_{\text{total}}$) is: $$ \vec{E}_{\text{total}} = \vec{E}_1 + \vec{E}_2 + \ldots + \vec{E}_n $$

5. Electric Field Lines

Electric field lines are a visual tool to represent electric fields. They originate from positive charges and terminate at negative charges. The density of these lines indicates the strength of the electric field in a region. Key properties include:

• They never intersect.
• The tangent to a field line at any point gives the direction of the electric field at that point.
• The number of lines is proportional to the magnitude of the charge.

6. Electric Field Due to Multiple Point Charges

When dealing with multiple point charges, calculating the electric field involves applying the superposition principle meticulously. For example, consider two point charges, $q_1$ and $q_2$, located at different points in space. The electric field at a point $P$ due to both charges is: $$ \vec{E}_P = \vec{E}_{P1} + \vec{E}_{P2} $$ where $\vec{E}_{P1} = k_e \frac{q_1}{r_1^2} \hat{r}_1$ and $\vec{E}_{P2} = k_e \frac{q_2}{r_2^2} \hat{r}_2$, with $\hat{r}_1$ and $\hat{r}_2$ being unit vectors in the direction from each charge to point $P$.

7. Electric Field Intensity and Potential

While the electric field provides information about the force per unit charge, the electric potential ($V$) relates to the potential energy per unit charge. For point charges, the relationship between electric field and potential is: $$ \vec{E} = -\nabla V $$ Understanding both concepts is crucial for solving problems related to electric fields and potentials.

8. Applications of Electric Fields Due to Point Charges

Electric fields due to point charges have numerous applications, including:

• Electrostatic Force Calculations: Determining the force between charges in various configurations.
• Electric Potential Energy: Calculating the work done in assembling a charge distribution.
• Capacitance: Understanding the storage of electric charge in capacitors.
• Electric Field Mapping: Visualizing field lines for analyzing force distributions.

9. Gauss's Law and Symmetry

Gauss's Law provides a powerful method for calculating electric fields when symmetry is present. For point charges, although Gauss's Law is universally applicable, it is particularly efficient in scenarios with spherical symmetry.

Gauss's Law states: $$ \oint \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0} $$ where $Q_{\text{enc}}$ is the total charge enclosed within a Gaussian surface and $\varepsilon_0$ is the vacuum permittivity.

10. Electric Field Calculations Examples

To solidify understanding, consider the following examples:

1. Single Point Charge: Calculate the electric field 2 meters away from a charge of $5 \times 10^{-6} \, \text{C}$.
   Solution: $$ E = k_e \frac{|q|}{r^2} = 8.988 \times 10^9 \times \frac{5 \times 10^{-6}}{(2)^2} = 8.988 \times 10^9 \times \frac{5 \times 10^{-6}}{4} = 1.1235 \times 10^4 \, \text{N/C} $$
2. Two Opposite Charges: Find the electric field at the midpoint between a $+3 \times 10^{-6} \, \text{C}$ and a $-3 \times 10^{-6} \, \text{C}$ charge separated by 4 meters.
   Solution: The magnitudes of the fields due to each charge are equal but directions are opposite. Thus, the fields add up: $$ E = 2 \times \left( k_e \frac{3 \times 10^{-6}}{(2)^2} \right) = 2 \times \left( 8.988 \times 10^9 \times \frac{3 \times 10^{-6}}{4} \right) = 1.3491 \times 10^5 \, \text{N/C} $$

Comparison Table

Summary and Key Takeaways