Electric Forces and Fields: Study Notes ch 23

Electric Forces and Fields

Three Charges: Electric Force

When multiple point charges are present, the net electric force on any one charge is the vector sum of the forces exerted by all other charges. This is a direct application of Coulomb's Law and the principle of superposition.

• Coulomb's Law: The magnitude of the force between two point charges is given by

$$ F = k_e \frac{|q_1 q_2|}{r^2}$$

• Superposition Principle: The net force on a charge is the vector sum of the forces from all other charges.

• Example: For three charges at specified coordinates, calculate the force on one by summing the vector contributions from the other two. Components must be resolved along x and y axes, and the resultant magnitude is found using the Pythagorean theorem.

The Field Model

The field model describes how source charges create an electric field \( \vec{E} \) at all points in space. A separate charge in this field experiences a force:

$$ \vec{F}_{\text{on q}} = q \vec{E}$$

• The force on a positive charge is in the direction of \( \vec{E} \).

• The force on a negative charge is opposite to \( \vec{E} \).

• Units: Electric field is measured in newtons per coulomb (N/C).

• The magnitude of the electric field is called the electric field strength.

Direction of Electric Force and Field

The direction of the electric force and field depends on the sign of the charges involved:

• Like charges repel; opposite charges attract.

• The direction of \( \vec{E} \) at a point is the direction of the force on a positive test charge placed at that point.

• For a positive source charge, \( \vec{E} \) points away; for a negative source charge, \( \vec{E} \) points toward the charge.

Magnitude of Electric Force and Field

• The force between two charges depends on the product of their charges and the inverse square of their separation.

• The electric field at a point is independent of the charge used to measure it.

The Electric Field of a Point Charge

The electric field due to a point charge is:

$$ \vec{E} = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2} \hat{r}$$

• \( \varepsilon_0 = 8.85 \times 10^{-12} \; \text{C}^2/\text{N} \cdot \text{m}^2 \) is the permittivity of free space.

• \( \hat{r} \) is the unit vector from the charge to the point of interest.

Sign of Charge and Field Direction

• If the electric field vectors point away from a charge, the charge is positive.

• If the electric field vectors point toward a charge, the charge is negative.

• The field's magnitude decreases with distance from the charge.

Chapter 23: The Electric Field

23.1 Electric Field Model

Electric fields can be calculated for various charge distributions:

• Point charge

• Line of charge

• Plane of charge

• Sphere of charge

These are idealized models; real objects may only approximate these shapes.

23.2 Electric Field of Multiple Point Charges

The net electric field at a point due to several point charges is the vector sum of the fields from each charge:

$$ \vec{E}_{\text{net}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + \ldots = \sum_i \vec{E}_i$$

Finding Zero Electric Field

There may be points where the electric fields from multiple charges cancel each other, resulting in a net field of zero. This occurs where the vector sum of the fields is zero, often between or outside the charges depending on their signs and magnitudes.

Electric Field from Two Charges

The net electric field at a point due to two charges is found by vector addition of the fields from each charge. The direction and magnitude depend on the relative positions and signs of the charges.

Electric Field from Three Charges

For three charges, the net field is again the vector sum. Symmetry can simplify calculations, especially if charges are arranged in regular geometric patterns.

23.2 Electric Field of a Dipole

An electric dipole consists of two equal and opposite charges separated by a small distance. The dipole moment is:

$$ \vec{p} = q \vec{s}$$

• Points from negative to positive charge.

• The field at a distance r along the axis of the dipole:

$$ E_{\text{dipole, axis}} = \frac{1}{2\pi\varepsilon_0} \frac{p}{r^3}$$

• Perpendicular to the axis:

$$ E_{\text{dipole, perp}} = \frac{1}{4\pi\varepsilon_0} \frac{p}{r^3}$$

Electric Field Lines

• Electric field lines are tangent to the electric field vector at every point.

• Lines start on positive charges and end on negative charges.

• Field lines never cross.

• Closer spacing indicates a stronger field.

Continuous Charge Distribution

For objects with charge distributed over a length or area:

• Linear charge density: \( \lambda = \frac{Q}{L} \)

• Surface charge density: \( \sigma = \frac{Q}{A} \)

• The total field is found by integrating the contributions from each infinitesimal charge element.

Semi-Circular Rod

For a semi-circular rod with charges +Q and -Q distributed along the top and bottom halves, the horizontal components of the electric field at the center cancel, leaving only the vertical component.

Electric Field Due to a Point Charge (Component Form)

For a charge q at (0, y0), the x-component of the field at (xP, 0) is:

$$ E_x = \frac{1}{4\pi\varepsilon_0} \frac{q x_P}{(x_P^2 + y_0^2)^{3/2}}$$

Electric Field Due to a Line of Charge

For a thin rod of length l and charge q on the y-axis, the x-component of the field at (xP, 0) is:

$$ E_x = \frac{1}{4\pi\varepsilon_0} \int_{-l/2}^{l/2} \frac{\lambda x_P}{(x_P^2 + y^2)^{3/2}} dy$$

Common Charge Distributions

Shape	Electric Field Magnitude
Line of charge	$E = \frac{1}{2\pi\varepsilon_0} \frac{\lambda}{r}$
Ring of charge	$E = \frac{1}{4\pi\varepsilon_0} \frac{qz}{(z^2 + R^2)^{3/2}}$
Plane of charge	$E = \frac{\sigma}{2\varepsilon_0}$

23.5 The Parallel-Plate Capacitor

A parallel-plate capacitor consists of two plates with equal and opposite charges. The electric field between the plates is:

$$ E_{\text{cap}} = \frac{Q}{\varepsilon_0 A}$$

• The field is uniform between the plates and points from the positive to the negative plate.

23.6 Motion of a Charged Particle in an Electric Field

• A positively charged particle accelerates in the direction of the electric field.

• A negatively charged particle accelerates opposite to the field.

• The force is given by $\vec{F} = q\vec{E}$ and the acceleration by $\vec{a} = \frac{q\vec{E}}{m}$.

23.7 Motion of a Dipole in an Electric Field

The torque on a dipole in an electric field is:

$$ \vec{\tau} = \vec{p} \times \vec{E}$$

• The magnitude is $\tau = pE \sin\theta$, where $\theta$ is the angle between $\vec{p}$ and $\vec{E}$.

• In a uniform field, the dipole rotates to align with the field but does not translate.

• In a non-uniform field, the dipole experiences both torque and net force, causing translation.

Summary Table: Key Equations

Concept	Equation
Coulomb's Law	$F = k_e \frac{|q_1 q_2|}{r^2}$
Electric Field (point charge)	$E = \frac{1}{4\pi\varepsilon_0} \frac{q}{r^2}$
Force on charge in field	$\vec{F} = q\vec{E}$
Dipole moment	$\vec{p} = q\vec{s}$
Torque on dipole	$\vec{\tau} = \vec{p} \times \vec{E}$

Additional info: These notes include both conceptual explanations and worked examples, as well as quiz-style questions and their solutions, to reinforce understanding of electric forces and fields.