🧲AP Physics 2 Unit 3 – Electric Force, Field, and Potential

Key Concepts and Definitions

• Electric charge is a fundamental property of matter that causes it to experience a force when placed in an electromagnetic field
• Electric field is a region around an electric charge in which an electric force is exerted on another charge
• Electric potential energy is the energy that is needed to move a charge against an electric field
• Electric potential (voltage) is the amount of electric potential energy per unit charge
• Coulomb's law states that the magnitude of the electrostatic force of attraction or repulsion between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them
• Conductors are materials that allow electric charges to flow freely through them (metals, graphite, salt water)
• Insulators are materials that do not allow electric charges to flow freely through them (rubber, plastic, glass)
• Capacitance is the ability of a system to store an electric charge

Fundamental Laws and Equations

• Coulomb's law: $F = k \frac{q_1 q_2}{r^2}$, where $F$ is the electric force, $k$ is Coulomb's constant ($8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$), $q_1$ and $q_2$ are the magnitudes of the charges, and $r$ is the distance between the charges
• Electric field strength: $E = \frac{F}{q}$, where $E$ is the electric field strength, $F$ is the electric force, and $q$ is the test charge
• Electric potential energy: $U = qV$, where $U$ is the electric potential energy, $q$ is the charge, and $V$ is the electric potential (voltage)
• Electric potential (voltage): $V = \frac{U}{q}$, where $V$ is the electric potential (voltage), $U$ is the electric potential energy, and $q$ is the charge
• Capacitance: $C = \frac{Q}{V}$, where $C$ is the capacitance, $Q$ is the charge stored, and $V$ is the voltage across the capacitor
  • Capacitance of a parallel plate capacitor: $C = \frac{\epsilon_0 A}{d}$, where $\epsilon_0$ is the permittivity of free space ($8.85 \times 10^{-12} \frac{F}{m}$), $A$ is the area of the plates, and $d$ is the distance between the plates
• Electric field of a point charge: $E = k \frac{q}{r^2}$, where $E$ is the electric field strength, $k$ is Coulomb's constant, $q$ is the magnitude of the point charge, and $r$ is the distance from the point charge

Electric Charge and Its Properties

• Electric charge is a property of matter that causes it to experience a force when placed in an electromagnetic field
• There are two types of electric charges: positive and negative
• Like charges repel each other, while unlike charges attract each other
• Electric charge is conserved, meaning that the total charge in a closed system remains constant
• Electric charge is quantized, meaning that it comes in discrete units (multiples of the elementary charge, $e = 1.602 \times 10^{-19}$ C)
  • Protons have a charge of $+e$, electrons have a charge of $-e$, and neutrons have no charge
• Electric charge can be transferred through conduction (direct contact) or induction (proximity to a charged object)
• The SI unit for electric charge is the coulomb (C)

Electric Force and Coulomb's Law

• Electric force is the force experienced by a charged particle in an electric field
• Coulomb's law states that the magnitude of the electrostatic force between two point charges is directly proportional to the product of the magnitudes of charges and inversely proportional to the square of the distance between them
  • $F = k \frac{q_1 q_2}{r^2}$, where $F$ is the electric force, $k$ is Coulomb's constant ($8.99 \times 10^9 \frac{N \cdot m^2}{C^2}$), $q_1$ and $q_2$ are the magnitudes of the charges, and $r$ is the distance between the charges
• The direction of the electric force depends on the signs of the charges
  • Like charges repel each other (force points away from each other)
  • Unlike charges attract each other (force points toward each other)
• The electric force is a conservative force, meaning that the work done by the force is independent of the path taken
• The electric force is a long-range force, meaning that it can act over large distances
• The electric force is much stronger than the gravitational force (about $10^{36}$ times stronger for two protons)

Electric Fields: Concept and Calculations

• An electric field is a region around an electric charge in which an electric force is exerted on another charge
• The electric field strength is defined as the force per unit charge: $E = \frac{F}{q}$, where $E$ is the electric field strength, $F$ is the electric force, and $q$ is the test charge
• The electric field is a vector quantity, with magnitude and direction
  • The direction of the electric field is the direction of the force on a positive test charge
• The electric field of a point charge is given by: $E = k \frac{q}{r^2}$, where $E$ is the electric field strength, $k$ is Coulomb's constant, $q$ is the magnitude of the point charge, and $r$ is the distance from the point charge
• The electric field of a collection of point charges can be found using the principle of superposition (adding the individual electric fields as vectors)
• The electric field inside a conductor is zero, as the charges redistribute themselves to cancel out the field
• The electric field near a charged conductor is perpendicular to the surface of the conductor

Electric Potential and Voltage

• Electric potential energy is the energy that is needed to move a charge against an electric field
  • $U = qV$, where $U$ is the electric potential energy, $q$ is the charge, and $V$ is the electric potential (voltage)
• Electric potential (voltage) is the amount of electric potential energy per unit charge
  • $V = \frac{U}{q}$, where $V$ is the electric potential (voltage), $U$ is the electric potential energy, and $q$ is the charge
• The electric potential is a scalar quantity, with only magnitude
• The electric potential difference (voltage) between two points is the work done per unit charge to move a positive test charge from one point to the other
• The electric potential of a point charge is given by: $V = k \frac{q}{r}$, where $V$ is the electric potential, $k$ is Coulomb's constant, $q$ is the magnitude of the point charge, and $r$ is the distance from the point charge
• The electric potential of a collection of point charges can be found using the principle of superposition (adding the individual electric potentials as scalars)
• The electric potential is always defined relative to a reference point (usually taken to be infinity, where $V = 0$)

Applications and Real-World Examples

• Capacitors are devices that store electric charge and energy in an electric field
  • Capacitors are used in various electronic circuits (filters, power supplies, memory devices)
• Van de Graaff generators use the principles of electric charge and electric fields to generate high voltages
  • Used in particle accelerators, X-ray machines, and electrostatic experiments
• Lightning is a natural example of electric discharge, where the electric potential difference between a cloud and the ground or another cloud is equalized through a sudden flow of electric charges
• Electrostatic precipitators use electric fields to remove pollutants (smoke, dust) from exhaust gases in industrial settings
• Xerography (photocopying) uses the principles of electric charge and electric fields to create images on paper
  • A charged photoconductor is selectively discharged by light, and the remaining charges attract toner particles which are then transferred to paper
• Cathode ray tubes (CRTs) in old television sets and computer monitors use electric fields to guide electron beams to create images on a phosphorescent screen
• Inkjet printers use electric fields to control the trajectory of charged ink droplets onto paper

Common Problems and Problem-Solving Strategies

• Identifying the type of problem (electric force, electric field, electric potential) and the relevant equations
• Drawing a clear diagram of the problem, including charges, distances, and directions
• Converting units to the appropriate SI units (meters, coulombs, volts)
• Applying the principle of superposition for multiple charges (adding electric fields as vectors, adding electric potentials as scalars)
• Using symmetry to simplify problems (e.g., the electric field at the center of a uniform ring of charge is zero)
• Applying Gauss's law to calculate the electric field of symmetric charge distributions (spheres, cylinders, planes)
  • Gauss's law: $\oint \vec{E} \cdot d\vec{A} = \frac{Q_{enc}}{\epsilon_0}$, where $\vec{E}$ is the electric field, $d\vec{A}$ is the area element, $Q_{enc}$ is the total charge enclosed by the surface, and $\epsilon_0$ is the permittivity of free space
• Using the work-energy principle to relate electric potential differences and changes in kinetic energy
  • $\Delta U = q \Delta V = \Delta KE$, where $\Delta U$ is the change in electric potential energy, $q$ is the charge, $\Delta V$ is the electric potential difference, and $\Delta KE$ is the change in kinetic energy
• Applying conservation of energy to solve problems involving the motion of charged particles in electric fields