【People's Education Press】High School Physics - Required Course, Volume 3: Characteristics of Electrostatic Work: Path Independence

In this section, we explore a fundamental mechanical property of electrostatic fields:path independence. Imagine you're facing a fluctuating energy map—electrostatic force acts like gravity, a mover that cares only about the outcome, not the journey.

Core Definitions and Principles

Uniform Electric Field: If the electric field strength is equal in magnitude and direction at every point in the field, it is called a uniform electric field. A typical model isa pair of parallel metal plates with equal but opposite charges placed very close together.
Work Formula: When moving charge $q$ in a uniform electric field $E$, the work done is $W_{AB} = F \cdot |AB| \cos \theta = qE d$. Here, $d$ is the distance between the initial and final positions along the direction of the electric field lines.
Path Independence Theorem: When moving a charge in a uniform electric field, the work done by the electrostatic force depends only on the starting and ending positions of the charge, not on the path taken.

Physical Analogy: Gravity vs Electrostatic Force

Electrostatic force, like gravity, is a "conservative force." This means that for any force whose work is independent of the path, we can introduce a corresponding energy concept (such as electric potential energy or gravitational potential energy) to describe the energy transformation caused by its work.

QUESTION 1

Which of the following statements correctly describes the nature of electrostatic work in a uniform electric field?

The work done by the electric force depends on the length of the charge’s trajectory

The work done by the electric force is non-zero when a charge moves around a closed loop

The work done by the electrostatic force depends only on the charge’s initial and final positions, not on the path taken

The work done by the electrostatic force satisfies $W = qEd$ only along straight-line paths

QUESTION 2

What type of electric field exists in the internal region (excluding edges) between two closely spaced parallel metal plates with equal but opposite charges?

Non-uniform electric field

Point charge electric field

Uniform Electric Field

Alternating electric field

QUESTION 3

In studies of microscopic particles, the electronvolt (eV) is commonly used as an energy unit. How many joules does 1 eV correspond to when an electron is accelerated through a 1 V potential difference?

$1.0 \text{ J}$

$1.602 \times 10^{-19} \text{ J}$

$9.11 \times 10^{-31} \text{ J}$

QUESTION 4

A positively charged small ball ($m = 1.0 \times 10^{-3} \text{ kg}, q = 2.0 \times 10^{-8} \text{ C}$) is suspended by a lightweight insulating string. In a horizontal uniform electric field pointing right, it reaches equilibrium with the string making a $30^\circ$ angle with the vertical. Find the electric field strength $E$.

$1.44 \times 10^5 \text{ N/C}$

$2.89 \times 10^5 \text{ N/C}$

$5.00 \times 10^5 \text{ N/C}$

QUESTION 5

If the electrostatic force does +10 J of work on a moving charge, how does the charge's electric potential energy change?

Increases by $10 \text{ J}$

Decreases by $10 \text{ J}$

Remains unchanged

Case Study: Static Discharge in Daily Life and Path-Independent Work

Exploring the energy transformations and path-independent characteristics behind static phenomena

Electrostatic force acts not only in lab settings like parallel plates, but also in our everyday lives—from spark discharges in dry seasons to charge movements in complex fields. Understanding its work characteristics unlocks the mystery of energy transformation.

1. During dry weather, you often get shocked when touching a metal doorknob after removing your outer clothing. Explain this phenomenon from the perspective of electrostatic work.

Answer:
This is a static discharge phenomenon. In dry weather, friction between clothing and body or inner layers generates and accumulates static charge (triboelectric charging). When your hand approaches a metal doorknob, a strong electric field forms between your body and the conductor, ionizing the air and causing sudden charge transfer (discharge). During this process, electrostatic force does positive work on the charge, rapidly converting accumulated electric potential energy into heat and light—resulting in the sensation of an electric shock.

2. As shown in Figure 10.1-5, in a uniform electric field $E = 60 \text{ N/C}$, there are points $A$, $B$, and $C$. $AB = 5 \text{ cm}$ (along the field direction), and $BC = 12 \text{ cm}$ (at a $60^\circ$ angle to the field direction). Calculate the total work done by the electrostatic force when moving a charge $q = 4 \times 10^{-8} \text{ C}$ from $A$ via $B$ to $C$. What if it were moved directly from $A$ to $C$?

Answer:
(1) Work from A to B: $W_{AB} = qE \cdot |AB| = 4 \times 10^{-8} \times 60 \times 0.05 = 1.2 \times 10^{-7} \text{ J}$. (2) Work from B to C: $W_{BC} = qE \cdot |BC| \cos 60^\circ = 4 \times 10^{-8} \times 60 \times 0.12 \times 0.5 = 1.44 \times 10^{-7} \text{ J}$. (3) Total work: $W_{ABC} = W_{AB} + W_{BC} = 2.64 \times 10^{-7} \text{ J}$. (4) Directly from A to C: Since electrostatic work is path-independent, the result matches the path via B exactly—still $2.64 \times 10^{-7} \text{ J}$.