Relationship Between Work and Electric Force

Introduction

Key Concepts

Electric Force

Electric force is the interaction between charged particles, governed by Coulomb's Law. It describes the force exerted by one charge on another and is pivotal in determining the behavior of charges in electric fields.

Coulomb's Law is mathematically expressed as: $$ F = k_e \frac{|q_1 q_2|}{r^2} $$ where:

• $F$ is the magnitude of the electric force between the charges.
• $k_e$ is Coulomb's constant ($8.988 \times 10^9 \, \text{N.m²/C²}$).
• $q_1$ and $q_2$ are the amounts of the two charges.
• $r$ is the distance between the centers of the two charges.

The force is attractive if the charges are of opposite signs and repulsive if they are of the same sign.

Work in Electric Fields

Work is a measure of energy transfer when a force causes the displacement of an object. In the context of electric fields, work is done when a charge moves within an electric potential.

The work ($W$) done by an electric force when a charge moves from point A to point B is given by: $$ W = \int_{A}^{B} \mathbf{F} \cdot d\mathbf{s} $$ where:

• $\mathbf{F}$ is the electric force.
• $d\mathbf{s}$ is the differential displacement vector.

If the electric force is constant and the motion is along the direction of the force, this simplifies to: $$ W = F \cdot d \cdot \cos(\theta) $$ where $\theta$ is the angle between the force and the displacement vectors.

Electric Potential Energy

Electric potential energy ($U$) is the energy stored in a system of charged particles due to their positions in an electric field. It is a scalar quantity and plays a crucial role in energy conservation within electric systems.

For a pair of point charges, the electric potential energy is given by: $$ U = k_e \frac{q_1 q_2}{r} $$ where the symbols represent the same quantities as in Coulomb's Law.

This equation indicates that the potential energy increases as charges are moved apart and decreases as they are brought closer together, depending on the nature of the charges.

Relationship Between Work and Electric Force

The work done by an electric force is directly related to the change in electric potential energy. When a charge moves in an electric field, the work performed by the electric force results in a change in the system's electric potential energy.

The relationship is mathematically expressed as: $$ W = -\Delta U = U_{\text{initial}} - U_{\text{final}} $$

This negative sign indicates that if work is done by the electric force, the electric potential energy of the system decreases, and vice versa.

Conservative Nature of Electric Forces

Electric forces are conservative forces, meaning the work done by these forces is path-independent and depends only on the initial and final positions. This property allows the definition of electric potential energy.

The conservative nature leads to the concept that the total mechanical energy (kinetic plus potential) in a system remains constant in the absence of non-conservative forces.

Electric Potential and Potential Difference

Electric potential ($V$) is defined as the electric potential energy per unit charge. It provides a measure of the potential energy landscape in which charges move.

Mathematically, it is expressed as: $$ V = \frac{U}{q} $$ where:

• $V$ is the electric potential.
• $U$ is the electric potential energy.
• $q$ is the charge.

The potential difference between two points ($\Delta V$) is the work done per unit charge when moving a charge between those points: $$ \Delta V = \frac{W}{q} $$

This concept is essential for understanding how electric circuits function and how energy is transferred within them.

Energy Conservation in Electric Systems

In electric systems, energy conservation dictates that the total energy remains constant unless acted upon by external forces. The work done by electric forces leads to transformations between kinetic and potential energy.

For example, when a charge accelerates due to an electric force, the decrease in electric potential energy results in an increase in the charge's kinetic energy, maintaining the conservation of total energy.

Examples Illustrating the Relationship

Consider two charges, $q_1$ and $q_2$, separated by a distance $r$. Moving $q_2$ a small distance $\Delta r$ towards $q_1$ involves doing work against the electric force if the charges are like charges.

The work done is: $$ W = F \cdot \Delta r = k_e \frac{q_1 q_2}{r^2} \cdot \Delta r $$

This work results in a change in electric potential energy: $$ \Delta U = U_{\text{final}} - U_{\text{initial}} = k_e \frac{q_1 q_2}{r - \Delta r} - k_e \frac{q_1 q_2}{r} $$

If $\Delta r$ is positive, indicating a decrease in separation, and the charges are like charges, the work done increases the system's potential energy.

Mathematical Derivation of Work-Energy Relationship

Starting from Coulomb's Law and the definition of work, we can derive the relationship between work and electric potential energy.

Given: $$ F = k_e \frac{q_1 q_2}{r^2} $$ and work: $$ W = \int_{r_i}^{r_f} F \, dr = \int_{r_i}^{r_f} k_e \frac{q_1 q_2}{r^2} dr = -k_e \frac{q_1 q_2}{r} \Big|_{r_i}^{r_f} = -\Delta U $$

Thus, the work done by the electric force is the negative change in electric potential energy.

Implications in Electric Circuits

In electric circuits, the relationship between work and electric force is manifested in the behavior of charges flowing through components like resistors, capacitors, and inductors. The work done by electric forces drives the movement of electrons, leading to current and the dissipation or storage of energy within the circuit elements.

Understanding this relationship is crucial for analyzing circuit behavior, energy efficiency, and the functioning of various electrical devices.

Comparison Table

Summary and Key Takeaways