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Physics II

Physics II



Module 1: Electric Charges, Forces, and Fields

Electric charge is a fundamental property of matter which determines how it interacts with electromagnetic fields. There are both positive and negative charges, with the same charges repeling and opposite charges attracting.
Electric charge is quantized
  • SI unit of electric charge is the coulomb (C)
  • Elementary charge is
    
Proton:

Neutron:

Electron:


Law of Conservation of Charge
Charge is conserved—the net charge of the universe or a local closed environment is constant. When we speak of charges “cancelling,” we are referring to the oppositely directly forces applied cancel for a net force of zero. The charge itself does not disappear.

An electric force is created by electric charges, acting without physical contact between the two objects, but its magnitude strengthening with proximity and the amount of electric charge.
Electrostatic Repulsion: Interacting objects with same sign of charge have a repulsive force
Electrostatic Attraction: Interacting objects with opposite sign of charge have an attractive force
*Magnitude is separate from type of charge (+/-)
Coulomb's Law
If two objects have electric charge, they exert an electric force on each other. The direction of the force vector is along the imaginary line joining the two objects, dictated by signs of charges involved. Its range is infinite.



Principle of Superposition:
When there's multiple source charges, the analysis can be repeated with a test charge: another point charge that is used to test the strength of the force provided by the source charges
Force doesn't necessarily point in same direction as the unit vector—may be opposite depending upon the signs of the source/test charge



Electromagnetic forces are much stronger than gravity—every other force studied in Physics I is electromagnetic in nature, as contact is an illusion due to the repulsion of outer electrons of substances.


Electrical Properties of Materials

In an atom, electrons surround a tiny nucelus of protons and neutrons in the form of a vast cloud of negative charge. The outermost valence electrons can wander from atom to atom, driving the movement of charge. A material's properties is dependent on the strength of the valence electron bonds.
Conductor: Electric current flows freely. Loosely bound conduction electrons.
Insulator: Current doesn't flow readily. Tightly bound valence electrons.
Semi-conductors: Determined by impurities, which can change conductivity through the number of loosely bound valence electrons.

Excess charge on an insulator stays in place and result in electrical attractive and repulsive forces, while excess charge placed on a conductor will instantly flow away.
Charging by Induction
When an electrically charged object is brought close to a conductor, it exerts an electric force on the conduction electrons, which flow towards the object.
While the conductor is still neutrally charged, it now has charge distribution, forming an electric dipole through inducing polarization.

When brought near an insulator or neutral substance, the distribution of charge in atoms/molecules changes. Due to the electric force decreasing with distance, the attraction of unlike charges is stronger than the relpulsion of like charges—neutral objects can be attracted to a charged object.

Static Electricity: Rubbing two neutral objects together can displace normally stationary electrons through friction, resulting in an imbalance of charge within the object. This results in the “shock” of a rapid flow of electrons to another object when a conductive path is available to equalize the charge.


Electric Fields

A field is a physical quantity whose value depends on (is a function of) position, relative to the source. It is a way of describing how an object influences the space around it, giving a value at every point in space.
  • We often use vector fields, which assign a magnitude and direction to each point, making a collection of an infinite number of distinct vectors across space.
A familiar field we understand is a gravitational field, which tells us how much force a mass will feel if it is there. However, the field exists even without the mass—fields separate the source from the effect

An electric field is the effect a charge has on the space around it—how much electric force would be applied per coulomb of charge.

The electric charge on an object alters the space around it in such a way where all other electrically charged objects experience an electric force as a result of being in the field.
E⃗=F⃗q,  NC=Vm\vec{E} = \frac{\vec{F}}{q}, \space \space \frac{N}{C} = \frac{V}{m}

E⃗:electric field at a pointF⃗:force on a small positive test chargeq:test charge\vec{E}: \text{electric field at a point} \\\vec{F}: \text{force on a small positive test charge} \\q: \text{test charge}

Follows superposition, as do forces: if multiple charges are present, the net electric field at a point is the vector sum.
E⃗net=∑iE⃗iE⃗net=E⃗1+E⃗2+E⃗3+...\vec{E}_{\text{net}} = \sum_i \vec{E}_i \\ \vec{E}_{\text{net}} = \vec{E}_1 + \vec{E}_2 + \vec{E}_3 + ...

For an electric field due to a point charge,
E⃗=kQr2r^\vec{E} = \frac{kQ}{r^2} \hat{r}

Continuous Charge Distributions:
When a charge is spread out amongst an object, break it into tiny sections,
, and add up the charges.
dE=kdqr2 E⃗=k∫dqr2r^dE = k\frac{dq}{r^2} \\\space \\\vec{E} = k\int\frac{dq}{r^2} \hat{r}



Geometrical Charge Distributions

Linear Charge Distribution: Charge per Length           λ=QL          dq=λ dL Area Charge Distribution: Charge per Area                 σ=QA          dq=σ dA Volume Charge Distribution: Charge per Volume        ρ=QV          dq=ρ dV\text{\small{Linear Charge Distribution: Charge per Length}} \space\space\space\space\space\space\space\space\space\space\space \lambda = \frac{Q}{L} \space\space\space\space\space\space\space\space\space\space dq = \lambda\space dL \\\space \\\text{\small{Area Charge Distribution: Charge per Area}} \space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space\space \sigma = \frac{Q}{A} \space\space\space\space\space\space\space\space\space\space dq = \sigma\space dA \\\space \\\text{\small{Volume Charge Distribution: Charge per Volume}} \space\space\space\space\space\space\space\space \rho = \frac{Q}{V} \space\space\space\space\space\space\space\space\space\space dq = \rho\space dV

The electric field shows the strength and direction of the electric force a positive test charge would feel at any point in space. By convention, all electric fields point away from positive source charges and toward negative charges.


Electric Field Lines / Diagrams

Allows a visualization of how the space is altered through geometrical sketches. Each arrow is a representation of the field vector at a point.
Vector Diagram

  • Magnitude: corresponds with length
  • Direction: radially away from source charge

Field Line Diagram

  • Direction: direction of field vector at point—field vectors are tangent to field lines
  • Magnitude/Strength: indicated by field line density, number of field lines per unit area
Drawing Field Line Diagrams:
  • Field lines originate on positive charges and terminate on negative charges. Number of field lines originating/terminating from a charge is proportional to charge magnitude.
  • In isolated charges, lines go to or from infinity.
  • Field lines can never cross—an electric field cannot point in two directions at a single point.
  • Lines are perpendicular to conductor surfaces.




Module 2: Gauss's Law

Electric Flux

The concept of flux describes how much of something goes through a given area—the dot product of a vector field with an area.
Electric flux is a measure of the number of electric field lines passing through an area. For uniform electric field through a flat area, it is the scalar product of the electric field and area vector.
  • In general, when field lines flow out of a closed surface, φ is positive, and when they flow into a surface, φ is negative.

For a non-flat surface, it can be divided into an infinitesimal small flat flux elements, finding the net flux by summing up each individual flux element (integration across the non-flat surface).
ΦE=E⃗⋅A⃗=EAcos⁡θ ΦE=∫SE⃗⋅dA⃗ΦE=∮SE⃗⋅dA⃗*Circle represents that surface isclosed and integration is happeningover entire surface. A portion of aclosed surface is treated as an opensurface subset\Phi_E = \vec{E} \cdot \vec{A} = EA\cos\theta \\\space \\\Phi_E = \int_S \vec{E} \cdot d\vec{A} \\\Phi_E = \oint_S \vec{E} \cdot d\vec{A} \\\scriptsize{\text{*Circle represents that surface is}} \\\scriptsize{\text{closed and integration is happening}} \\\scriptsize{\text{over entire surface. A portion of a}} \\\scriptsize{\text{closed surface is treated as an open}} \\\scriptsize{\text{surface subset}}

An area vector of a flat surface has a magnitude equal to area,
, and a direction that is normal (perpendicular) to the surface. For an open surface, any direction can be used as long as it is consistent. If a surface is closed, then the vector points from the inside to the outside.


Gauss's Law

If a closed surface does not have any charge inside of it where an electric field can terminate, then the electric flux through the surface is zero. However, if there is charge inside the enclosed volume, this vlue is equal to the total charge enclosed divided by the permittivity of free space constant.

Φ=∮SE⃗⋅dA⃗=Qencε0\Phi =\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

If
 has the same magnitude everywhere on the Gaussian surface and
 is parallel to the outward normal, then

∮E dA=E∮dA=EA\oint E\space dA = E \oint dA = EA



*Permittivity of free space/vacuum constant:





Module 3: Electric Potential


Electric Potential Energy:


ΔV=ΔPEq=−Wq=−F⃗⋅d⃗q=−E⃗⋅d⃗\Delta V = \frac{\Delta PE}{q} = \frac{-W}{q} = \frac{-\vec{F} \cdot \vec{d}}{q} = -\vec{E} \cdot \vec{d}



The Electron-Volt

The energy given to a fundamental charge accelerated through a potential difference of 1 V.










Equations and Concepts Reference

ELECTRICITY

Charges and Constants

Elementary Charge:

Coulomb's Constant:

Vacuum Permittivity:


Charge Distribution

Linear:

Surface (Area):
or

Volume:
or


Electric Fields

The effect a charge has on the space around it: the strength and direction a positive test charge would experience. Points toward negative electric charges.
E=k∣Q∣r2          E⃗=F⃗q          E⃗=∑E⃗iE = k\frac{|Q|}{r^2} \space\space\space\space\space\space\space\space\space\space \vec{E}= \frac{\vec{F}}{q} \space\space\space\space\space\space\space\space\space\space \vec{E} = \sum{\vec{E}_i}


Electric Forces

Coulomb's Law determines the magnitude between two point charges. Direction is determined by the charge signs—opposites attract.
F=kq1q2r2     or     F⃗=kq1q2r2r^F = k\frac{q_1 q_2}{r^2} \space\space\space\space\space\text{or}\space\space\space\space\space \vec{F} = k\frac{q_1 q_2}{r^2} \hat{r}


Electric Flux

The amount of electric field lines going through an area. An area vector is normal to the surface, = A.
Φ=E⃗⋅A⃗=EAcosθ        Φ=∮SE⃗⋅dA⃗=Qencε0\Phi = \vec{E} \cdot \vec{A} = EAcos\theta \space\space\space\space\space\space\space\space \Phi =\oint_S \vec{E} \cdot d\vec{A} = \frac{Q_{\text{enc}}}{\varepsilon_0}

The integral often simplifies and isn't needed:
  • Point charge with uniformly charged sphere
  • Spherical shell—thick uses
    
  • Infinite line charge or plane, between plates

Electric Potential

The stored energy due to the position of charges per unit charge at a point.
Scalar—the sign of Q is included. Add algebraically.
V=kQr          V=k∑QiriV = k\frac{Q}{r} \space\space\space\space\space\space\space\space\space\space V = k\sum\frac{Q_i}{r_i}

points in the direction of decreasing potential.
ΔV=−∫E⃗⋅dl⃗          ΔV=−E⋅d E⃗=−∇V   ⇒   Ex=−dVdx,  etc..\Delta V = - \int \vec{E} \cdot d\vec{l} \space\space\space\space\space\space\space\space\space\space \Delta V = -E \cdot d \\\space \\\vec{E} = -\nabla V \space\space\space \Rightarrow \space\space\space E_x = -\frac{dV}{dx}, \space\space\text{etc..}


Conductors and Insulators

In electrostatic equilibrium, excess charge in a conductor object resides on the outside.
  •      inside
  •      outside = point charge at the center.

For an insulator, the charge can be distributed throughout the volume or surface—charge distrubutions are used and integrated unless high symmetry allows for Gauss's Law to be used.

Work and Energy

Potential Energy:

Work:

Acceleration: Positive to lower V, Negative to higher V


*Forces and fields both follow superposition—the net value is the vector sum.