Electric Charges and Fields

Introduction to Electric Charges and Fields

• Explores electric force as a non-contact force similar to gravitational force.
• Focuses on electric charge and its related phenomena.

Electric Charge

What is Electric Charge?

• Electric charge is acquired when two different materials are rubbed together, causing attraction.
• It is a property of matter causing force exertion in an electromagnetic field.
• There are two types of charges: positive and negative with like charges repelling and unlike charges attracting.
• The SI unit for electric charge is the coulomb (C).
• Electric charge is quantized, existing in discrete packets that are integer multiples of the elementary charge (e).
  • e = $1.602192 \times 10^{-19} C$
• An electron has a charge of -e, while a proton has a charge of +e.
• Charge is conserved; the total charge within an isolated system remains constant.

Conductors and Insulators

• Materials are classified by their ability to conduct electricity.
  • Conductors allow electric charge to move freely (e.g., metals, human body).
  • Insulators restrict the movement of electric charge (e.g., glass, plastic, wood).

Charging by Induction

• Induction is charging an object without direct contact.
• A charged object is brought near a neutral object, causing charge redistribution.
• Grounding the neutral object allows charge flow between the object and the ground.
• Removing the ground and then the charged object leaves the neutral object charged.

Basic Properties of Electric Charge

Additivity of Charges

• The total charge of a system is the algebraic sum of individual charges.
  • If charges are $q_1, q_2, q_3,..., q_n$, the total charge Q is: $Q = q_1 + q_2 + q_3 +... + q_n$.

Charge is Conserved

• The total charge in an isolated system remains constant.
• Charge is neither created nor destroyed, only transferred.

Quantization of Charge

• Electric charge is an integral multiple of the elementary charge e.
  • $q = ne$, where n is an integer (0, ±1, ±2, ±3,...).

Coulomb's Law

Coulomb's Law

• The force between two point charges is directly proportional to the product of their magnitudes and inversely proportional to the square of the distance between them.
  • $F = k \frac{|q_1 q_2|}{r^2}$
• Where:
  • F is the electrostatic force
  • $q_1$ and $q_2$ are the magnitudes of the charges
  • r is the distance between the charges
  • k is Coulomb's constant
    • $k = \frac{1}{4 \pi \epsilon_0} = 8.9875 \times 10^9 Nm^2C^{-2} \approx 9 \times 10^9 Nm^2C^{-2}$
    • $\epsilon_0$ is the permittivity of free space, equalling $8.854 \times 10^{-12} C^2N^{-1}m^{-2}$.

Vector Form of Coulomb's Law

• The force on charge $q_1$ due to $q_2$ is: $\overrightarrow{F}{12} = k \frac{q_1 q_2}{r^2{12}} \hat{r}_{12}$.
  • $\overrightarrow{F}_{12}$ is the force on $q_1$ due to $q_2$.
  • $\hat{r}_{12}$ is the unit vector from $q_2$ to $q_1$.
  • $r_{12}$ is the distance between $q_1$ and $q_2$.

Forces between Multiple Charges

• The total force on a charge due to multiple charges is the vector sum of individual forces. $\overrightarrow{F}1 = \overrightarrow{F}{12} + \overrightarrow{F}{13} +... + \overrightarrow{F}{1n}$

Electric Field

Electric Field

• An electric field is a region around a charged object where force is exerted on other charges.
• Defined as $\overrightarrow{E} = \frac{\overrightarrow{F}}{q}$, where $\overrightarrow{F}$ is the force on a small positive test charge $q$.
• The SI unit of electric field is N/C or V/m.

Electric Field due to a Point Charge

• Given by $\overrightarrow{E} = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2} \hat{r}$, where $\hat{r}$ is the unit vector from the charge to the point of calculation.

Electric Field due to a System of Charges

• The electric field is the vector sum of fields from individual charges.
  • $\overrightarrow{E} = \overrightarrow{E}_1 + \overrightarrow{E}_2 +... + \overrightarrow{E}_n$

Electric Field Lines

Electric Field Lines

• Imaginary lines representing the direction and strength of an electric field.

Properties of Electric Field Lines

• Start from positive charges and end at negative charges.
• The number of lines is proportional to the charge magnitude.
• Tangent to a line indicates the electric field direction.
• Field lines never intersect and the density shows field strength.

Electric Flux

Area Vector

• The area vector $\overrightarrow{\Delta S}$ has a magnitude equal to the area and a direction normal to the surface.

Electric Flux

• Measures the number of electric field lines through a surface with $\Delta \phi = \overrightarrow{E} \cdot \overrightarrow{\Delta S} = E \Delta S \cos \theta$.
  • $\Delta \phi$ is the electric flux through the surface $\Delta S$.
  • $\overrightarrow{E}$ is the electric field.
  • $\overrightarrow{\Delta S}$ is the area vector.
  • $\theta$ is the angle between $\overrightarrow{E}$ and $\overrightarrow{\Delta S}$.
• For a closed surface, the total electric flux is: $\phi = \oint \overrightarrow{E} \cdot d\overrightarrow{S}$.

Electric Dipole

Electric Dipole

• Pair of equal but opposite charges separated by a small distance $2a$.
• The center of the dipole is the midpoint of the line joining the charges.

Dipole Moment

• $\overrightarrow{p}$ equals $q(2a)$, directed from the negative to positive charge. The SI unit is Coulomb-meter (Cm).

Electric Field of a Dipole

• At a point on the axial line: $E = \frac{1}{4 \pi \epsilon_0} \frac{2pr}{(r^2 - a^2)^2}$. If $r >> a$, then $E \approx \frac{1}{4 \pi \epsilon_0} \frac{2p}{r^3}$.
• At a point on the equatorial line: $E = \frac{1}{4 \pi \epsilon_0} \frac{p}{(r^2 + a^2)^{3/2}}$. If $r >> a$, then $E \approx \frac{1}{4 \pi \epsilon_0} \frac{p}{r^3}$.

Dipole in a Uniform External Field

• Experiences a torque: $\overrightarrow{\tau} = \overrightarrow{p} \times \overrightarrow{E}$ or $\tau = p E \sin \theta$, with $\theta$ as the angle between $\overrightarrow{p}$ and $\overrightarrow{E}$.
• Potential energy $U = - \overrightarrow{p} \cdot \overrightarrow{E}$ or $U = -pE \cos \theta$.

Continuous Charge Distribution

Continuous Charge Distribution

• Distribution of electric charge continuously over a region.

Types of Charge Distribution

• Linear charge density ($\lambda$): Charge per unit length, $\lambda = \frac{dQ}{dl}$.
• Surface charge density ($\sigma$): Charge per unit area, $\sigma = \frac{dQ}{dS}$.
• Volume charge density ($\rho$): Charge per unit volume, $\rho = \frac{dQ}{dV}$.

Electric Field due to a Continuous Charge Distribution

• Given by $\overrightarrow{E} = \frac{1}{4 \pi \epsilon_0} \int \frac{dQ}{r^2} \hat{r}$, where $dQ$ is the charge element and $r$ is the distance to the point of calculation.

Gauss's Law

Gauss's Law

• The total electric flux through a closed surface equals the enclosed charge divided by $\epsilon_0$: $\phi = \oint \overrightarrow{E} \cdot d\overrightarrow{S} = \frac{Q_{encl}}{\epsilon_0}$.
  • $\phi$ signifies the electric flux through a surface.
  • $\overrightarrow{E}$ stands for the electric field.
  • $d\overrightarrow{S}$ is the area vector.
  • $Q_{encl}$ is the total enclosed charge.
  • $\epsilon_0$ is the permittivity of free space.

Applications of Gauss's Law

• Electric field due to an infinitely long straight wire: $E = \frac{\lambda}{2 \pi \epsilon_0 r}$.
• Electric field due to a uniformly charged infinite plane sheet: $E = \frac{\sigma}{2 \epsilon_0}$.
• Electric field due to a uniformly charged thin spherical shell:
  • For $r < R$: $E = 0$.
  • For $r \geq R$: $E = \frac{1}{4 \pi \epsilon_0} \frac{Q}{r^2}$.