Relationship Between Work Done by Electric Force and Change in Electric Potential Energy
In electromagnetism, the relationship between the work done by the electric force and the change in electric potential energy is a fundamental principle that serves as a direct manifestation of the law of conservation of energy within electric fields. This connection dictates how energy transforms between kinetic and potential forms as charges move through space. To understand this dynamic, we must first establish the nature of the electric field, define the concept of work in this context, and then derive the precise mathematical link to potential energy.
The Nature of the Electric Field and Conservative Forces
The electric field, denoted by $\mathbf{E}$, is a vector field defined at every point in space. It describes the force per unit positive charge that would be exerted on a test charge placed at that location, expressed as $\mathbf{F} = q\mathbf{E}$. A critical characteristic of electrostatic fields is that they are conservative fields.
A conservative field implies that the net work done by the field on a particle moving along any closed path is zero:
[
\oint \mathbf{F}\cdot d\mathbf{l} = 0
]
Because the work done is path-independent, we can define a scalar quantity called electric potential energy ($U$) for a charge $q$ within the field. This property allows us to treat electric potential energy similarly to gravitational potential energy, where the system's state is determined solely by position rather than the trajectory taken to reach it.
Defining Work in an Electric Context
In mechanics, work ($W$) is defined as the line integral of the force component along the direction of displacement. For a charge $q$ moving from point $A$ to point $B$ under the influence of the electric field, the work done by the electric force ($W_e$) is calculated as:
[
W_{e} = \int_{A}^{B} \mathbf{F}\cdot d\mathbf{l} = \int_{A}^{B} q\mathbf{E}\cdot d\mathbf{l} = q\int_{A}^{B}\mathbf{E}\cdot d\mathbf{l}
]
Here, $A$ and $B$ represent the initial and final positions of the particle. The dot product $\mathbf{E}\cdot d\mathbf{l}$ ensures that only the component of the electric field parallel to the displacement contributes to the work.
Electric Potential and Potential Energy
To simplify the analysis of energy changes, physicists introduce the concept of electric potential ($\varphi$), which is the potential energy per unit charge. It is defined relative to a reference point (usually infinity where potential is zero) as:
[
\varphi(\mathbf{r}) = -\int_{\infty}^{\mathbf{r}} \mathbf{E}\cdot d\mathbf{l}
]
The electric potential energy ($U$) of a charge $q$ at a specific point is then given by:
[
U = q\varphi
]
Since $\varphi$ is a scalar function of position, the change in potential energy ($\Delta U$) depends only on the starting and ending points, not the path taken. This scalar nature makes it significantly easier to calculate energy changes compared to evaluating complex vector integrals along arbitrary curves.
The Core Relationship: Work and Potential Energy Change
The most profound insight in electrostatics is the direct inverse relationship between the work done by the electric field and the change in potential energy. Because the electric force is conservative, the work done by the field as a charge moves from $A$ to $B$ is exactly equal to the negative of the change in potential energy:
[
W_{e} = -\Delta U = -(U_{B} - U_{A})
]
This equation encapsulates the physical intuition:
- If the electric field does positive work on a charge, the charge's potential energy decreases.
- If the electric field does negative work (meaning an external agent must do work against the field), the charge's potential energy increases.
Mathematically, substituting the definitions of work and potential energy yields:
[
q\int_{A}^{B}\mathbf{E}\cdot d\mathbf{l} = -q\bigl[\varphi(B)-\varphi(A)\bigr]
]
Dividing by $q$ (assuming $q \neq 0$) leads to the standard definition of electric potential difference (or voltage):
[
\Delta\varphi = \varphi(B)-\varphi(A) = -\int_{A}^{B}\mathbf{E}\cdot d\mathbf{l}
]
This confirms that the potential difference between two points is the negative of the work done per unit charge to move a test charge between those points.
Conservation of Energy in Electric Systems
The relationship $W_e = -\Delta U$ is the bridge to the broader principle of conservation of energy. In a system where only the electric force acts on a charged particle, the total mechanical energy remains constant. The work done by the electric force equals the change in the particle's kinetic energy ($K$):
[
W_{e} = \Delta K = K_{B} - K_{A}
]
Combining this with the potential energy relationship, we arrive at the conservation law:
[
K_{A} + U_{A} = K_{B} + U_{B}
]
This equation states that any decrease in potential energy must result in an equivalent increase in kinetic energy, and vice versa. This framework allows us to solve complex motion problems without needing to know the specific forces acting at every instant, provided we know the initial and final positions.
Practical Applications and Examples
1. Motion of a Charge Near a Point Charge
Consider a positive charge $q$ moving in the field of another positive charge $Q$. As $q$ moves closer to $Q$ (from distance $r_1$ to $r_2$ where $r_2 < r_1$), the electric force (repulsive) does positive work. Consequently, the potential energy decreases, and the kinetic energy of $q$ increases, accelerating it toward the source.
2. Uniform Electric Fields
In a uniform electric field $\mathbf{E} = E\hat{x}$, moving a charge $q$ a distance $d$ along the field lines results in a potential difference $\Delta\varphi = -Ed$. The work done is $W_e = qEd$. If the charge moves with the field, it loses potential energy and gains speed; if it moves against the field, it gains potential energy at the expense of kinetic energy (or requires external work).
3. Capacitor Discharge
In a parallel plate capacitor, storing charge creates an electric field and stores energy as potential energy. When the capacitor discharges, the electric field does work on the charges as they flow through the circuit. The loss in electric potential energy ($\Delta U$) is converted into other forms, such as thermal energy (heat) or light, strictly adhering to the principle that the work done by the field equals the reduction in stored energy.
Conclusion
The interplay between electric work and potential energy change forms the backbone of electrostatic analysis. By recognizing that the electric field is conservative, we establish that work depends only on endpoints, allowing us to define a scalar potential energy function. The equation $W_e = -\Delta U$ provides a powerful tool for analyzing energy transformations, ensuring that in isolated systems, the sum of kinetic and potential energy remains invariant. Whether calculating the trajectory of a particle in a cathode ray tube or determining the energy output of a battery, this fundamental relationship remains the guiding principle for understanding electromagnetic systems.