Electric Charge, Coulomb’s Law, Conductors, Insulators, & Polarization – Physics 1200 Lecture 01 Study Notes

Electric Charge and Electric Forces

Observational Data

Electric forces are a fundamental aspect of physics, observed as unique interactions between certain objects. These forces can be generated by rubbing two objects together, sometimes resulting in a spark or a 'zap' that dissipates the force or changes its sign. The strength of the force decreases with distance, following an inverse-square law similar to gravity, but unlike gravity, electric forces can be attractive, repulsive, or zero depending on the types of charges involved.

• Electric force acts along the line joining two objects.

• Can be generated by friction (rubbing objects).

• Force magnitude decreases with distance: $F \propto \frac{1}{r^2}$.

• Electric forces can be attractive or repulsive, unlike gravity which is always attractive.

• Interactions depend on the types of charges present.

Example: If object A attracts B but repels C, then B and C attract each other. If A feels no force from D, then B and C also feel no force from D.

Electric Charge

The property responsible for electric forces is called electric charge. There are two types of charge: positive and negative. Ben Franklin coined these terms. The fundamental law is that like charges repel and unlike charges attract. Charge is always conserved in a closed system, and experiments (such as the Millikan oil drop experiment) have shown that charge is quantized—it exists in discrete amounts.

• Electric charge is a conserved quantity.

• Two types: positive and negative.

• Like charges repel; unlike charges attract.

• Charge is quantized: $q = n e$, where $n$ is an integer and $e$ is the elementary charge.

Example: The Millikan oil drop experiment demonstrated the quantization of electric charge.

Charge and Electric Forces

Because both attractive and repulsive forces exist, it is concluded that there are two types of charge. The direction of the force depends on the sign of the charges involved.

• Positive charges ($+$) repel other positive charges.

• Negative charges ($-$) repel other negative charges.

• Positive and negative charges attract each other.

Example: Diagrams show force vectors between charges of the same and opposite signs.

Coulomb’s Law

Mathematical Formulation

Coulomb’s Law describes the force between two point charges. The magnitude and direction of the force depend on the values and signs of the charges, as well as the distance between them.

• Coulomb’s Law:

$\vec{F}_{12} = k \frac{q_1 q_2}{r_{12}^2} \hat{r}_{12}$

• $\vec{F}_{12}$: Electric force exerted by $q_1$ on $q_2$

• $q_1$, $q_2$: Point charges

• $r_{12}$: Distance between charges

• $\hat{r}_{12}$: Unit vector from $q_1$ to $q_2$

• $k = \frac{1}{4\pi \varepsilon_0}$, where $\varepsilon_0 = 8.85 \times 10^{-12} \text{C}^2/\text{N} \cdot \text{m}^2$ is the permittivity of free space

• $k \approx 9.0 \times 10^9 \text{N} \cdot \text{m}^2/\text{C}^2$

Example: If charges have the same sign, the force is repulsive; if opposite, the force is attractive.

Fundamental Physical Properties of Charge

Units and Quantities

The SI unit of charge is the coulomb (C). The charge on a single electron is $-1.602 \times 10^{-19}$ C. All observed charges are integer multiples of the elementary charge $e$.

Object	Charge (C)	Mass (kg)	Charge/Mass (C/kg)
Electron	−1.60 × 10−19	9.11 × 10−31	1.76 × 1011
Proton	+1.60 × 10−19	1.67 × 10−27	9.58 × 107
Neutron	0	1.67 × 10−27	0
Carbon atom	0	20.0 × 10−27	0
Person shuffling on a carpet	~10−10	~102	~10−12

Example: The charge on a proton is $+e$, and on an electron is $-e$.

Additional info: The charge on a person shuffling on a carpet is much larger than the charge on a single electron, but still small in everyday terms.

Vector Addition of Electric Forces

Superposition Principle

The net electric force on a charge is the vector sum of all individual forces exerted by other charges. This is known as the principle of superposition.

• Forces are vectors: sum components in each direction.

• For $N$ charges, the net force on charge $i$ is:

$\vec{F}_i = \vec{F}_{i1} + \vec{F}_{i2} + \vec{F}_{i3} + \ldots$

• Resolve each force into x, y, (and z) components:

$F_{ix} = F_{i1x} + F_{i2x} + \ldots$ $F_{iy} = F_{i1y} + F_{i2y} + \ldots$ $F_{iz} = F_{i1z} + F_{i2z} + \ldots$

Example: For multiple charges, calculate each force vector and sum their components.

Unit Vectors

A unit vector defines direction in space and has magnitude 1. Any non-zero vector can be expressed as a product of its magnitude and a unit vector.

• Unit vector: $|\hat{r}| = 1$

• Constructed as $\hat{r} = \vec{r}/|\vec{r}|$

• Used to specify direction in Coulomb’s Law

Example: The vector $\vec{r} = r \hat{r}$, where $r$ is the magnitude and $\hat{r}$ is the unit vector.

Solving Vector Force Problems

Algorithm for Vector Addition

To solve problems involving vector addition of forces, follow these steps:

1. Draw a diagram with all vectors.

2. Choose axes to simplify calculations (use symmetry if possible).

3. Resolve each vector into x- and y-components:

$A_{ix} = A_i \cos \theta_i$ $A_{iy} = A_i \sin \theta_i$

4. Add components to get total sums:

$A_x = \sum_i A_{ix}$ $A_y = \sum_i A_{iy}$

5. Calculate the angle of the resultant vector:

$\tan \theta = \frac{A_y}{A_x}$

Additional info: Extension to three dimensions follows similarly by including z-components.

Conductors, Insulators, and Polarization

Classes of Materials

Materials respond differently to electric fields and forces, depending on whether they are conductors or insulators.

• Conductors: Allow charge to move freely. Charge placed on a conductor redistributes to make the internal electric field zero.

• Charge can be placed directly or transferred by induction.

• Excess charge can be 'run to ground' via a conducting path.

• Insulators: Charge remains fixed in position. Must be placed directly on the insulator.

Example: Attaching a wire to ground provides a path for excess charge to leave a conductor.

Polarization and Attraction

Polarization is the rearrangement of charges within a material in response to an external electric field or charge. In conductors, free electrons move, creating regions of positive and negative charge. In insulators, polarization is much weaker due to limited internal charge motion.

• In conductors, polarization leads to strong attractive forces toward external charges.

• In insulators, polarization effects are weak.

Example: Bringing a negatively charged rod near an uncharged metal ball causes electrons in the ball to move away from the rod, leaving positive charge closer to the rod. This separation of charge is polarization, resulting in an attractive force.

Practice and Example Problems

Example Problem 1.1

Four charges, each with $|q| = 10^{-7}$ C, are placed at the corners of a square 1 m on a side. Find the force on a $+10^{-7}$ C charge at the center of the square.

• Use Coulomb’s Law to calculate the force from each corner charge.

• Sum the vector forces to find the net force at the center.

Example Problem 1.2

Four charges, each with $|q|$, are placed at the corners of a square of side $a$. Calculate the total force on the charge in the upper right corner.

• Calculate individual vector forces from each of the other charges using Coulomb’s Law.

• Perform vector addition to find the net force on the corner charge.

Additional info: These problems illustrate the use of vector addition and symmetry in calculating electric forces in multi-charge systems.