Electrostatics
2.1 Electric Charge
Focus Question: What causes an object to have a net electric charge?
Bohr Model of the Atom: The nucleus contains protons (positive charge) and neutrons (no charge). The nucleus's charge depends only on the number of protons.
Protons are about 2000 times more massive than electrons; almost all of an atom's mass is in the nucleus.
Electrons exist outside the nucleus, orbiting it. Electrons have very little mass but have an equal and opposite charge to a proton.
Electric Charge: A property of matter, can be positive or negative. Negative charge is associated with electrons, positive with protons.
Conservation of Charge: Charge can be transferred between objects but cannot be created or destroyed.
Quantity of Charge: Measured in Coulombs (C).
Elementary charge: e = 1.6 \times 10^{-19} C, the magnitude of charge of an electron (-) or proton (+).
Charge is often given in micro Coulombs (\mu C) because a Coulomb is a large unit.
Quantization of Charge: Charge is quantized, meaning it exists in integral multiples of a fundamental unit.
Total negative charge: Q_{net} = -ne, where n is the number of extra electrons.
Total positive charge: Q_{net} = +ne, where n is the number of missing electrons.
Charge is due to the flow of electrons, as they are free to leave an atom, unlike protons which are bound by the strong nuclear force. A negative charge indicates excess electrons, while a positive charge indicates a deficit.
Conductors: Materials with free electrons that can move easily, facilitating charge transfer through electron flow.
High conductivity implies low resistivity.
Charge accumulates on the surface of a conductor.
Most metals are good conductors.
Insulators: Materials where electrons are tightly bound to the nucleus, impeding charge transfer.
Examples: glass, rubber, and plastic.
Polarization: A conductor brought near a charged object will polarize fully. Charges separate due to electrostatic force.
Insulator Polarization: Insulators can also exhibit charge polarization at the atomic level, leading to attraction to charged objects.
The electroscope: Used to detect charge. It consists of two foil leaves connected to a conducting rod and a metal knob, all housed in a jar. When charged, the leaves repel and diverge; greater charge magnitude results in greater divergence.
Charging by Conduction: Both insulators and conductors can be charged through direct contact.
A charged rod touching an uncharged electroscope transfers charge. A negatively charged rod transfers electrons; a positively charged rod attracts electrons from the electroscope.
The electroscope retains a charge of the same sign as the rod after the rod is removed.
Charging by Induction: Conductors are charged without direct contact.
Bring a charged object near the electroscope knob without touching it.
Grounding: Connect the electroscope to the earth, allowing electron flow to neutralize charge.
Break the ground connection, trapping the induced charge.
Remove the charged object; the charge redistributes evenly, leaving the electroscope with a charge opposite the rod.
2.2 Electric Force
Focus Question: What is Coulomb’s Law?
Fundamental Forces of Nature:
Gravitational Force: Attraction between masses, infinite range, caused by masses bending spacetime.
Electromagnetic Force: Described by Coulomb’s law, infinite range, mediated by photons.
Weak Nuclear Force: Responsible for beta decay, short range, mediated by W and Z bosons.
Strong Nuclear Force: Holds nuclei together, very short range, mediated by gluons.
For evenly distributed charge, the charge can be considered to be concentrated at the object's center.
The net charge is the difference between the number of electrons and protons.
Properties of Electrostatic Force:
Vector quantity along the line joining the particles.
Opposite charges attract, like charges repel.
Directly proportional to the product of the charges and follows the inverse square law for distance.
Coulomb's Law:
F = k\frac{|Q1||Q2|}{r^2} k = 8.99 \times 10^9 Nm^2/C^2
k = \frac{1}{4\pi\epsilon_0}
\epsilon_0 = permittivity of free space = 8.85 \times 10^{-12} C^2/Nm^2: measure of resistance to electric field formation in a vacuum.
Superposition: The net electrostatic force on a charge is the vector sum of forces due to other charges.
2.3 Electric Field
Focus Question: What is a field force?
A field force acts at a distance (e.g., gravity, electrostatic force, magnetic force).
Any electric charge creates an electric field around it. Electric field is defined based on the force a positive test charge would experience.
Force on a test charge near charge Q: F = \frac{kQq}{r^2}
Electric field strength: E = \frac{F}{q} \rightarrow E = \frac{kQ}{r^2} (Units: N/C).
Electric field is a vector quantity.
A positive charge moves in the direction of the electric field; a negative charge moves opposite to it.
Electric Field Lines: Vector field indicating the direction a positive test charge would experience.
Lines go towards negative charges, away from positive charges.
Positive charges move along the field lines; negative charges move opposite to them.
Stronger fields have more field lines.
2.4 Electric Potential
Focus Question: What does electric potential measure?
Potential relates to potential energy. Potential energy is the potential to do work.
A positive charge has higher potential near other positive charges.
General definition: an object has higher potential at locations where forces would cause it to move away.
Relationship between work and potential energy: W = -\Delta U \rightarrow \Delta U = -Fr
Electric Potential Energy:
Between like charges: U = k\frac{Qq}{r}. Positive since a positive test charge has potential to move away.
Between opposite charges: U = -k\frac{Qq}{r}. Negative since a positive test charge is "stuck".
General formula: U = Fr = k \frac{Qq}{r^2} (r) \rightarrow U = k\frac{Qq}{r}
Electric Potential:
Defined as the electrical potential energy per unit charge: V = \frac{U}{q}
\Delta U = Uf - Ui = q(Vf - Vi)
Units: J/C, Volts (V).
Potential difference between two points:
\Delta V = VB - VA = \frac{Work_{A \rightarrow B}}{q}
Scalar quantity, but can be positive or negative.
2.5 Capacitors
Focus Question: How is electrical energy stored in a capacitor?
Electric Field due to a charged plate: E = \frac{\sigma}{2\epsilon0} = \frac{Q}{2A\epsilon0} (Q = charge, A = area, \epsilon_0 = permittivity).
The electric field is independent of the distance from the plate (if the plate is large relative to the distance).
Capacitance: Measure of a capacitor's ability to store charge when a potential difference is applied: C = \frac{Q}{V}
Parallel-plate capacitor: C = \frac{\epsilon_0 A}{d} (A = area, d = distance).
Electric field between capacitor plates: E = \frac{V}{d}
More surface area means more capacitance.
Greater plate separation means less capacitance.
Energy Stored in a Capacitor: U = \frac{1}{2} QV = \frac{1}{2} CV^2
In a capacitor, it’s important for the space between the plates to be non-conductive. Capacitors can use air in the gap between the plates, but air can sometimes allow the flow of electricity.
Dielectric Breakdown: Charge jumps the gap between the plates of a capacitor