Electric Field Fundamentals: Coulomb's Law, Field Lines, and Dipoles with Walter Lewin

Overview

In this lecture Walter Lewin introduces the electric field concept as the force per unit charge. He revisits Coulomb's law, emphasizing that the field depends on source charges and not on the test charge, and defines both the direction and magnitude of E as a property of space around charges.

Key visual tools include vector arrows and field lines. Lewin explains that field vectors point in the direction a positive test charge would experience force and that field strength is represented by arrow length or line density. He also covers how multiple charges combine via the superposition principle and how dipole configurations differ from single charges.

In-lecture demonstrations

Demonstrations with an electroscope and a charged rod illustrate induction and dipole formation, while field lines and seeds visualizations offer intuitive insight into dipole fields and radial symmetry. The talk emphasizes that field lines are not trajectories but guides for the direction of force on charges, and it previews how dipole fields fall off as 1/r^3 at large distances.

Introduction to Electric Field

Walter Lewin begins by defining the electric field E at a location as the force on a test charge q divided by q, yielding a quantity that does not depend on q. He revisits Coulomb's law, F = k Q q / r^2, and explains that the field created by a source charge Q is felt by any test charge, with the sign of Q determining the field direction. The field is a vector quantity with E pointing away from positive charges and toward negative charges. By convention, the field direction corresponds to the force on a positive test charge.

In this framework, the field around a charge is independent of the test charge and can be visualized with field vectors and lines. Lewin notes that the unit for E in SI units is newtons per coulomb, though we often omit the unit in drawings.

Graphical Representations of the Field

The lecture moves from single charges to more complex configurations. For a positive charge, arrows emanate outward, larger near the charge and shrinking with distance, illustrating the inverse-square dependence. For a negative charge, arrows point inward. Lewin emphasizes spherical symmetry for point charges and shows how the field vectors indicate the force direction on a positive test charge. He also points out that the same configuration is seen from far away as if it were a single effective charge equal to the net source charge.

Lewin then introduces the idea of multiple charges, such as Q1, Q2, Q3, and demonstrates how the net field at a point is the vector sum E = E1 + E2 + E3 + .... This is the superposition principle in action, which allows the field to be calculated for any charge arrangement by summing each charge’s contribution.

Superposition and Far-Field Behavior

With several charges, the overall field is the vector sum of each charge’s contribution. If you know the E-field at a location, you can predict the force on any test charge placed there. Lewin emphasizes the consistency of this principle with experiments, even if the underlying microscopic arrangement seems complex. The idea that field strength can be visualized through the density of field lines is introduced, with closer lines indicating a stronger field.

Two-charge examples illustrate how the field can point in different directions depending on distance. For instance, a +3 and a -1 produce a field that, when far away, behaves like a +2 charge, but near the -1 it can be strongly directed toward the negative charge. This leads to the prediction that there exists a point where the field vanishes, a topic assigned as an exercise for students to explore numerically in a given configuration.

Dipoles and Field Lines

The discussion then moves to the dipole, a pair of charges with equal magnitude and opposite sign. The field far from a dipole falls off faster than 1/r^2, specifically as 1/r^3, making zero-field points nonexistent at large distances. The field around a dipole is highly symmetric near the charges, with lines radiating outward from the positive charge and converging toward the negative charge, and field lines between the charges indicating the direction of force on a positive test charge.

Lewin contrasts field lines with particle trajectories, noting that field lines indicate instantaneous force directions, not the actual path a moving charge would take in a curved field, except in ideal straight-field cases where a charge would follow the field line if released from rest.

Field Lines as a Visualization Tool

Field lines are introduced as a systematic way to visualize fields; their density conveys field strength, while their curvature reveals the local field direction. The speaker notes that lines are a conceptual tool rather than physical particles, and explains field configurations around multiple charges through line patterns. He uses analogies like air blowing from a positive charge and suction toward a negative charge to provide intuition for how lines emanate and terminate.

Experimental Demonstrations and Induction

The lecture culminates with dynamic demonstrations. Lewin shows how to induce a dipole in conductors using a charged rod, then separate the conductors to fix the dipole in place. An electroscope demonstrates polarity through leaf deflection, and a later experiment with grass seeds in oil shows seeds orienting along the field direction, visually confirming field line patterns around charges and dipoles. He also presents a simple toy experiment with two positively charged spheres to illustrate the formation and rotation of a dipole in an external field, and discusses how dipoles align with field lines when placed in a field.

Applications and Polarization

The concept of dipoles is linked to real-world phenomena such as induced polarization in atoms and molecules when placed in external fields, which underpins the study of dielectrics. Lewin highlights that dipoles are ubiquitous in physics and chemistry, and that understanding their behavior is fundamental to more advanced topics like polarization, dielectric constants, and material properties in external fields.

Overall, the lecture emphasizes that the electric field is a powerful, charge-independent representation of the environment created by charges, enabling precise prediction of forces on test charges and providing a rich visualization through vectors and field lines. It also foreshadows broader topics in electromagnetism and applications to matter, molecules, and materials.