Week 1 Lec. 3

Electric Field: Definition and Mapping

Test Charge and Electric Field Identification

The electric field is a fundamental concept in electromagnetism, representing the force per unit charge exerted on a test charge at a given point in space. To map an electric field, a positive test charge is placed at various points, and the force experienced by the charge is measured. The direction and magnitude of this force define the electric field at each location.

• Test Charge: A hypothetical charge (usually positive) used to probe the electric field without disturbing the source charges.

• Electric Field Vector: Points in the direction of the force experienced by a positive test charge.

• Formula: $\vec{E} = \frac{\vec{F}_0}{q_0}$, where $\vec{F}_0$ is the force on the test charge $q_0$.

• Units: Newtons per Coulomb (N/C).

Electric Field Lines and Charge Configurations

Visualizing Electric Fields

Electric field lines are graphical representations that show the direction and strength of the electric field. The tangent to a field line at any point gives the direction of the electric field vector at that point. Field lines originate from positive charges and terminate at negative charges.

• Field Line Properties:

  • Lines never cross.

  • Density of lines indicates field strength.

  • Direction is away from positive and toward negative charges.

• Common Configurations:

  • Single point charge: Radial field lines.

  • Dipole: Lines curve from positive to negative charge.

  • Two like charges: Lines repel and curve outward.

Electric Field of a Point Charge

Mathematical Expression and Example

The electric field produced by a point charge is given by Coulomb's law. The field radiates outward from a positive charge and inward toward a negative charge. The magnitude of the electric field at a distance $r$ from a point charge $Q$ is:

• Formula: $E = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2}$

• Vector Form: $\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r}$, where $\hat{r}$ is the unit vector from the source charge to the field point.

• Example: The electric field at the position of an electron orbiting a proton in a hydrogen atom (radius $0.053$ nm) is $E = 5.1 \times 10^{11}$ N/C, directed outward from the proton.

Superposition Principle: Electric Field from Multiple Charges

Vector Addition of Fields

When multiple charges are present, the total electric field at a point is the vector sum of the fields produced by each charge. This is known as the superposition principle.

• Formula: $\vec{E}_{\text{total}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \ldots$

• Each field: $\vec{E}_i = \frac{1}{4\pi\varepsilon_0} \frac{Q_i}{r_i^2} \hat{r}_i$

• Example: For two charges $q_1$ and $q_2$, the field at point $P$ is the sum of $\vec{E}_1$ and $\vec{E}_2$.

Uniform Electric Fields: Parallel Plate Capacitor

Characteristics and Applications

A uniform electric field is one in which the field strength and direction are constant throughout a region. This occurs, for example, between the plates of a parallel plate capacitor.

• Field Direction: From the positive plate to the negative plate.

• Field Strength: $E = \frac{V}{d}$, where $V$ is the potential difference and $d$ is the plate separation.

• Applications: Used in capacitors and particle accelerators.

Charge Density and Continuous Charge Distributions

Types of Charge Density

Charge density describes how charge is distributed over a region. There are three main types:

• Linear charge density ($\lambda$): $\lambda = \frac{Q}{L}$ (C/m)

• Surface charge density ($\sigma$): $\sigma = \frac{Q}{A}$ (C/m2)

• Volume charge density ($\rho$): $\rho = \frac{Q}{V}$ (C/m3)

Electric Field from Continuous Distributions

To find the electric field from a continuous charge distribution, divide the distribution into infinitesimal elements and sum their contributions:

• Formula: $\vec{E} = \frac{1}{4\pi\varepsilon_0} \int \frac{dQ}{r^2} \hat{r}$

• Examples:

  • Field from a charged rod, ring, or disk.

  • Use symmetry and integration to simplify calculations.

Electric Dipoles: Force and Torque in an Electric Field

Translational and Rotational Effects

An electric dipole consists of two equal and opposite charges separated by a distance $d$. In an electric field, a dipole experiences both a net force and a torque.

• Dipole Moment: $\vec{p} = q \vec{d}$

• Torque: $\vec{\tau} = \vec{p} \times \vec{E}$

• Force: $\vec{F} = q \vec{E}$ for each charge

• Motion: The dipole tends to align with the electric field direction.

Summary Table: Key Concepts

Concept	Definition/Formula	Example/Application
Electric Field	$\vec{E} = \frac{\vec{F}_0}{q_0}$	Force on test charge
Point Charge Field	$\vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{Q}{r^2} \hat{r}$	Field around proton
Superposition	$\vec{E}_{\text{total}} = \sum \vec{E}_i$	Multiple charges
Uniform Field	$E = \frac{V}{d}$	Parallel plate capacitor
Charge Density	$\lambda, \sigma, \rho$	Rod, disk, ring
Dipole Moment	$\vec{p} = q \vec{d}$	Torque in field

Additional info:

• Atomic structure and polarization are briefly referenced but not detailed in these notes.

• Worked examples for rods, rings, and disks are summarized; full integration steps can be found in standard physics textbooks.