Glossary

5.5 Calculating Electric Fields of Charge Distributions

2 min read•june 24, 2024

Electric charges come in two flavors: discrete and continuous. Discrete charges are individual points, like electrons. Continuous charges spread over lines, surfaces, or volumes, like a charged wire or sphere. Understanding both types is key to grasping electric fields.

Calculating electric fields from continuous charge distributions involves dividing them into tiny elements. We then use ###'s_law_0### to find each element's contribution and integrate over the whole distribution. This method works for line, surface, and volume charges.

Continuous Charge Distributions and Their Electric Fields

Discrete vs continuous charges

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  • Discrete charges consist of individual point charges (electrons, protons, ions)
    • Each charge is a multiple of the e=1.602×10−19e = 1.602 \times 10^{-19} C
  • Continuous charge distributions have charges distributed continuously over a line, surface, or volume (charged wire, sheet, sphere)
    • Charge quantization still applies, but the distribution can be approximated as continuous for mathematical convenience

Types of continuous charge distributions

  • Line charges distribute charge along a line or curve (one-dimensional)
    • λ\lambda is the charge per unit length λ=dQ/dl\lambda = dQ/dl
    • Charged wire is an example of a
  • Surface charges distribute charge over a surface or area (two-dimensional)
    • σ\sigma is the charge per unit area σ=dQ/dA\sigma = dQ/dA
    • Charged sheet or plate is an example of a
  • Volume charges distribute charge throughout a volume (three-dimensional)
    • ρ\rho is the charge per unit volume ρ=dQ/dV\rho = dQ/dV
    • Charged sphere or cube is an example of a volume charge

Electric fields from charge distributions

  • General approach divides the into infinitesimal elements
    1. Calculate the electric field contribution from each element using 's law
    2. Integrate the contributions over the entire
  • Line charge electric field at point PP due to a line charge with linear charge density λ\lambda:
    • dE⃗=14πε0λdlr2r^d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\lambda dl}{r^2} \hat{r}
    • Integrate along the line: E⃗=∫dE⃗=14πε0∫λdlr2r^\vec{E} = \int d\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{\lambda dl}{r^2} \hat{r}
    • ε0\varepsilon_0 is the
  • Surface charge electric field at point PP due to a surface charge with σ\sigma:
    • dE⃗=14πε0σdAr2r^d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\sigma dA}{r^2} \hat{r}
    • Integrate over the surface: E⃗=∬dE⃗=14πε0∬σdAr2r^\vec{E} = \iint d\vec{E} = \frac{1}{4\pi\varepsilon_0} \iint \frac{\sigma dA}{r^2} \hat{r}
  • Volume charge electric field at point PP due to a volume charge with ρ\rho:
    • dE⃗=14πε0ρdVr2r^d\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{\rho dV}{r^2} \hat{r}
    • Integrate over the volume: E⃗=∭dE⃗=14πε0∭ρdVr2r^\vec{E} = \iiint d\vec{E} = \frac{1}{4\pi\varepsilon_0} \iiint \frac{\rho dV}{r^2} \hat{r}
  • deals with stationary electric charges and their fields
  • : a pair of equal and opposite charges separated by a small distance
  • : a measure of the electric field passing through a given surface
  • : a mathematical operation that describes the outward flux of a from a given point

Key Terms to Review (40)

Charge Distribution: Charge distribution refers to the spatial arrangement and density of electric charges within a system or object. It is a fundamental concept in electrostatics that describes how electric charges are distributed and how this distribution influences the resulting electric fields and potentials.
Charles-Augustin de Coulomb: Charles-Augustin de Coulomb was an 18th century French physicist who is best known for his pioneering work in the field of electrostatics. His research and discoveries laid the foundation for our understanding of the fundamental laws governing electric charges and the forces they exert on one another.
Continuous charge distribution: A continuous charge distribution is a model where the electric charge is spread over a region of space rather than being confined to discrete points. This concept is used to calculate the electric field produced by objects with uniformly distributed charges.
Coulomb: A coulomb (C) is the SI unit of electric charge, representing the amount of charge transported by a constant current of one ampere in one second. One coulomb is equivalent to approximately $6.242 \times 10^{18}$ elementary charges.
Coulomb: The coulomb (symbol: C) is the SI unit of electric charge, named after the French physicist Charles-Augustin de Coulomb. It is a fundamental quantity that describes the amount of electric charge and is used extensively in the study of electric phenomena across various physics topics.
Coulomb's law: Coulomb's law describes the force between two charged objects, stating that the force is directly proportional to the product of the magnitudes of the charges and inversely proportional to the square of the distance between them. This principle is crucial for understanding interactions between electric charges, influencing how charges behave in different materials, and shaping the concept of electric fields.
Cylindrical symmetry: Cylindrical symmetry occurs when an object or system remains invariant under rotations about a central axis. It is commonly used in the context of electric fields and charge distributions.
Differential Element: A differential element is an infinitesimally small portion or segment of a larger object or system that is used to analyze and describe the behavior or properties of that system. It is a fundamental concept in calculus and is employed in various fields, including physics, engineering, and mathematics, to study continuous phenomena and derive mathematical expressions that govern the behavior of these systems.
Divergence: Divergence is a mathematical concept that describes the density of the outward flux of a vector field from an infinitesimal volume around a given point. It quantifies the density of the outward flux of a vector field from an infinitesimal volume around a given point. Divergence is a crucial concept in the study of electric fields, as it provides a way to analyze the behavior of electric fields and their sources.
Electric charge: Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field. It comes in two types: positive and negative.
Electric Charge: Electric charge is a fundamental property of matter that describes the ability of a particle or object to experience and generate electromagnetic forces. It is the source of all electromagnetic phenomena and is the basis for understanding the behavior of electric fields and electric currents.
Electric dipole: An electric dipole consists of two equal and opposite charges separated by a small distance. It creates an electric field and has a dipole moment, which is a vector quantity pointing from the negative to the positive charge.
Electric Dipole: An electric dipole is a pair of equal and opposite electric charges separated by a small distance. It is a fundamental concept in electrostatics that describes the electric field and potential created by a pair of equal but opposite charges.
Electric Flux: Electric flux is a measure of the total electric field passing through a given surface. It represents the number of electric field lines passing perpendicularly through a surface, and is a key concept in understanding the behavior of electric fields and charges.
Electric potential: Electric potential is the amount of electric potential energy per unit charge at a specific point in an electric field. It is measured in volts (V).
Electric Potential: Electric potential, also known as electrostatic potential, is a scalar quantity that represents the amount of work done per unit charge in moving a test charge from an infinite distance to a specific point in an electric field. It is a measure of the potential energy per unit charge at a given location within an electric field.
Electrostatics: Electrostatics is the branch of physics that studies electric charges at rest. It involves understanding the forces, fields, and potentials associated with static electric charges.
Electrostatics: Electrostatics is the study of electric fields and charges at rest. It encompasses the principles and laws governing the behavior of stationary electric charges and the electric fields they produce. This field of physics is foundational to understanding the interactions between charged particles and the properties of electric fields.
Elementary Charge: The elementary charge is the smallest known unit of electric charge, carried by a single electron or proton. It serves as the fundamental unit of electric charge in many physics concepts and equations.
Equipotential surfaces: Equipotential surfaces are hypothetical surfaces where the electric potential is constant throughout. This means that any point on a given equipotential surface has the same electric potential energy per unit charge, which implies that no work is done when moving a charge along this surface. Understanding equipotential surfaces helps clarify how electric fields interact with charged objects and their distributions.
Field Lines: Field lines are visual representations of electric fields, indicating the direction and strength of the field generated by charged objects. These lines emerge from positive charges and terminate at negative charges, showing how a positive test charge would move in the presence of the electric field. The density of the lines reflects the strength of the electric field; closer lines indicate a stronger field, while lines that are further apart represent a weaker field.
Gauss's Law: Gauss's law is a fundamental principle in electromagnetism that relates the electric flux through a closed surface to the total electric charge enclosed within that surface. It provides a powerful tool for calculating the electric field produced by various charge distributions.
Gaussian surface: A Gaussian surface is an imaginary closed surface used in Gauss's Law to calculate the flux of an electric field. The choice of Gaussian surface simplifies the calculation of electric fields due to symmetric charge distributions.
Gaussian Surface: A Gaussian surface is an imaginary, closed surface used in electrostatics to apply Gauss's law and calculate the electric field. It is a powerful tool for analyzing the electric field of charge distributions without having to solve complex integrals.
Infinite plane: An infinite plane is an idealized two-dimensional surface that extends infinitely in all directions. It is often used to simplify the calculation of electric fields and other physical properties.
Integration: Integration is a fundamental concept in calculus that describes the process of finding the area under a curve or the accumulation of a quantity over an interval. It is the inverse operation of differentiation, which involves finding the rate of change of a function.
Line Charge: A line charge is a one-dimensional distribution of electric charge that extends infinitely in one direction. It is a theoretical model used to study the electric fields and fluxes generated by charge distributions that can be approximated as a line rather than a point or a volume.
Linear Charge Density: Linear charge density is defined as the amount of electric charge per unit length along a charged line or distribution. It is typically represented by the symbol $$\lambda$$ and is crucial for calculating electric fields produced by charged wires or filaments, as well as for understanding how charge distributions interact with electric fields and potentials.
Permittivity of Free Space: Permittivity of free space is a fundamental physical constant that measures the ability of a vacuum to permit electric field lines. It plays a crucial role in electrostatics, affecting the strength of electric fields and the behavior of charge distributions in free space.
Point Charge: A point charge is an idealized model of an electric charge that is concentrated at a single point in space, with no physical size or dimensions. This concept simplifies the analysis of electric fields and forces, allowing for easier calculations and a clearer understanding of how electric charges interact with one another and produce electric fields.
Superposition Principle: The superposition principle states that the net effect of multiple sources or influences acting on a system is the sum of their individual effects. This principle is fundamental in understanding various physical phenomena, particularly in the fields of electricity, magnetism, and wave mechanics.
Surface Charge: Surface charge refers to the electrical charge that develops on the surface of an object or material. This charge can be positive, negative, or neutral, and it plays a crucial role in understanding the behavior of electric fields and the interactions between charged particles and surfaces.
Surface charge density: Surface charge density, denoted as $\sigma$, is the amount of electric charge per unit area on a surface. It is measured in coulombs per square meter ($C/m^2$).
Surface Charge Density: Surface charge density is the amount of electric charge per unit area on the surface of a charged object. It is an important concept in understanding the behavior of electric fields and electric potentials in various contexts, including charged distributions, conductors in electrostatic equilibrium, and the calculation of electric potential.
Symmetry: Symmetry is the quality of being made up of exactly similar parts facing each other or around an axis, center, or edge. It is a fundamental concept in physics that describes the invariance of a system under certain transformations, such as translations, rotations, or reflections.
Vector field: A vector field is a map that assigns a vector to every point in space. In the context of electric fields, it represents the direction and magnitude of the electric force experienced by a positive test charge at each point.
Vector Field: A vector field is a function that assigns a vector to every point in a specified region of space. It is a mathematical construct that describes the magnitude and direction of a quantity, such as a force or a velocity, at every point in a given space.
Volt/Meter: The volt per meter (V/m) is the unit used to measure the strength or magnitude of an electric field. It represents the potential difference, or voltage, per unit distance, which is a fundamental concept in understanding the behavior and effects of electric fields.
Volume charge density: Volume charge density is a measure of the amount of electric charge per unit volume in a region of space. It is denoted by the symbol $\rho$ and typically expressed in units of coulombs per cubic meter (C/m^3).
Volume Charge Density: Volume charge density is a measure of the electric charge per unit volume within a given region of space. It is a fundamental concept in the study of electromagnetism and is crucial for understanding the behavior of electric fields and electric potential in various charge distributions.
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