Undergrad Physics Class Summary

Atom and Charge

• Atoms consist of:

  • Protons (positive charge)

  • Neutrons (neutral charge)

  • Electrons (negative charge)

• Like charges repel, and opposite charges attract.

• Objects gain charge by gaining or losing electrons.

Coulomb's Law

• Describes the force between two charged objects:

  • $F = k \frac{Q<em>1 Q</em>2}{r^2}$

  • $F$ is the force.

  • $k$ is Coulomb's constant.

  • $Q<em>1$ and $Q</em>2$ are the charges of the objects.

  • $r$ is the distance between the objects.

• The force is proportional to the charge on each object. Doubling the charge on one object doubles the force.

• The force is inversely proportional to the square of the distance between the objects. Doubling the distance reduces the force by a factor of four.

• Similar to gravity but much stronger; the electric force is about $10^{20}$ times stronger than gravity.

  • Gravity equation: $F = G \frac{m<em>1 m</em>2}{r^2}$, where $G$ is the gravitational constant and $m<em>1$ and $m</em>2$ are the masses.

Electric Field

• The force that would be exerted on a positive test charge (+1) at a given point in space.

• Magnitude: $E = k \frac{Q}{r^2}$

• Direction is given by a unit vector $\hat{r}$ pointing from the charge to the point in space.

• The electric field from multiple charges is the vector sum of the electric fields from each charge.

Electric Field from Continuous Charge Distributions

• Break the object into a bunch of tiny charges $dQ$.

• Then, integrate the electric field from each of those charges.

  • $E = \int k \frac{dQ}{r^2}$

• Differential charge $dQ$ can be expressed in terms of charge density:

  • Charge density: $\lambda = \frac{Q}{L}$, where $Q$ is the total charge and $L$ is the length.

  • $dQ = \lambda dx$

• To solve the integral for the electric field, use the Pythagorean theorem to find the radius $r$, and find the $\hat{r}$ vector.

Electric Flux

• A measure of how much of a vector field flows through a given area.

• Defined as the dot product of the vector field and the area vector.

• For a constant electric field through a flat area:

  • $\Phi = E \cdot A = EA \cos(\theta)$

  • $\Phi$ is the electric flux.

  • $E$ is the electric field.

  • $A$ is the area vector.

  • $\theta$ is the angle between the electric field and the area vector.

• For a more complicated surface, break it up into tiny areas $dA$ and integrate:

  • $\Phi = \int E \cdot dA$

Gauss's Law

• Relates the electric flux through a closed surface to the charge enclosed by the surface.

• $\Phi = \frac{Q<em>{enc}}{\epsilon</em>0}$

  • $\Phi$ is the electric flux through the closed surface.

  • $Q_{enc}$ is the charge enclosed by the surface.

  • $\epsilon_0$ is the permittivity of free space.

• The closed surface is called a Gaussian surface.

• Gauss's law works for any closed surface, not just a sphere.

• Charges outside the surface do not contribute to the net flux.

Example: Electric Field from a Sheet of Charge

• Use a cylindrical Gaussian surface.

• The electric field is constant and perpendicular to the surface of the cylinder.

• The left side of Gauss's Law simplifies to $E*A$, where $A$ is the area of the end of the cylinder.

• The right-hand side simplifies to the charge density times the area enclosed. The charge density is the charge per unit area $\sigma = Q/A$.

• Solving for the electric field:

  • $E = \frac{\sigma}{2\epsilon_0}$

Voltage

• Electric potential energy per unit charge.

• $V = \frac{U}{q}$

  • $V$ is the voltage.

  • $U$ is the electric potential energy.

  • $q$ is the charge.

Work and Potential Energy

• The work done by a force is the force times the distance:

  • $W = F \cdot d = Fd \cos(\theta)$

  • $W$ is the work done.

  • $F$ is the force.

  • $d$ is the distance.

  • $\theta$ is the angle between the force and the distance.

• Potential energy is created when you do negative work.

• The electric potential energy is:

  • $U = -\int F \cdot ds$

  • $U$ is the potential energy.

  • $F$ is the force.

  • $ds$ is the change in distance.

Electric Potential Energy between Two Charges

• $U = k \frac{Q<em>1 Q</em>2}{r}$

Voltage and Electric Field

• Voltage is the negative accumulation of the electric field across some distance:

  • $V = - \int E \cdot dl$

  • $E$ is the electric field.

  • $dl$ is the length.

• If the electric field and the change in distance are perpendicular, then there is no change in voltage.

• Equipotential lines are lines perpendicular to the electric field, all of which are at the same voltage.

Batteries

• A 9V battery has 9 volts of potential energy between the two ends.

Relationship between Force, Electric Field, Voltage, and Potential Energy

• Force and electric field are related by:

  • $F = qE$

• Voltage and potential energy are related by:

  • $U = qV$

• Electric field and voltage are related by:

  • $V = - \int E \cdot dl$

• Force and potential energy are related by:

  • $U = - \int F \cdot dl$

Current

• The flow of electric charge.

• $I = \frac{\Delta Q}{\Delta t}$

  • $I$ is the current.

  • $\Delta Q$ is the change in charge.

  • $\Delta t$ is the change in time.

• Geometric definition:

  • $I = nqAv$

  • $n$ is the charge carrier density.

  • $q$ is the charge on each carrier.

  • $A$ is the cross-sectional area of the wire.

  • $v$ is the velocity of the charge carriers (drift velocity).

• Drift velocity is very slow, around 0.2 mm/s.

• When you turn on a light switch, it's almost instant because charges are already there so the energy is transferred quickly (like a conga line).

Resistance

• The opposition to the flow of electric current.

• $R$ is the resistance.

• $V = IR$ (Ohm's law)

  • $V$ is the voltage.

  • $I$ is the current.

• Geometric definition:

  • $R = \rho \frac{L}{A}$

  • $\rho$ is the resistivity.

  • $L$ is the length.

  • $A$ is the cross-sectional area.

Resistors in Series

• The effective resistance is the sum of the individual resistances.

• $R<em>{eff} = R</em>1 + R_2 + …$

Resistors in Parallel

• The inverse of the effective resistance is the sum of the inverses of the individual resistances.

• $\frac{1}{R<em>{eff}} = \frac{1}{R</em>1} + \frac{1}{R_2} + …$

Power

• The rate at which energy is transferred.

• $P = \frac{\Delta E}{\Delta t} = IV = I^2R = \frac{V^2}{R}$

  • $P$ is the power.

  • $\Delta E$ is the change in energy.

  • $\Delta t$ is the change in time.

  • $I$ is the current.

  • $V$ is the voltage.

  • $R$ is the resistance.

• A low resistance will get a big spike in energy.

Kirchhoff's Laws

• Used to analyze circuits that cannot be simplified using series and parallel combinations.

• Kirchhoff's Current Law (KCL):

  • The total current entering a junction must equal the total current leaving the junction.

  • Current in = Current out

• Kirchhoff's Voltage Law (KVL):

  • The sum of the voltage drops around any closed loop in a circuit must equal zero.

  • $\sum V = 0$

Capacitors

• A component that stores electrical energy.

• Consists of two conductive metal sheets connected to a voltage source.

• The charge builds up on the plates until the voltage across the plates is equal to the voltage of the source.

• Capacitance is defined as:

  • $C = \frac{Q}{V}$

  • $C$ is the capacitance.

  • $Q$ is the charge.

  • $V$ is the voltage.

Charging a Capacitor

• The charge on the capacitor over time can be described by:

  • $Q(t) = V * C * (1 - e^{-\frac{t}{RC}})$

Discharging a Capacitor

• The charge on the capacitor decays over time:

  • $V(t) = V_0 * (e^{-\frac{t}{RC}})$

  • $Q(t) = Q_0 * (e^{-\frac{t}{RC}})$

  where V0 is the initial voltage and Q0 is the initial charge.

Capacitors in Parallel

• The effective capacitance is the sum of the individual capacitances.

• $C<em>{eff} = C</em>1 + C_2 + …$

Capacitors in Series

• The inverse of the effective capacitance is the sum of the inverses of the individual capacitances.

• $\frac{1}{C<em>{eff}} = \frac{1}{C</em>1} + \frac{1}{C_2} + …$

Energy of a Capacitor

• The energy stored in a capacitor can be expressed as:

  • $E = \frac{1}{2}CV^2 = \frac{1}{2}QV = \frac{1}{2}\frac{Q^2}{C}$

Dielectrics in Capacitors

• An insulating material that increases the effective charge a capacitor can hold.

• The insulator (dielectric) weakens the electric field by a factor $\kappa$.

• Replaces the constant from $\epsilon<em>0$ in the capacitance to $\kappa * \epsilon</em>0$

Alternating Current (AC) Circuits

• In an AC circuit, the voltage source is a sine wave.
  $V(t) = V_0 \sin(\omega t)$

  • $V_0$ is the peak voltage.

  • $\omega$ is the angular frequency.

  Remember that Voltage still euqals current times resistance

Root Mean Square (RMS)

• The RMS voltage is the effective voltage of an AC circuit.

• $V<em>{rms} = \sqrt{\frac{1}{T} \int</em>0^T V(t)^2 dt} = \frac{V_0}{\sqrt{2}}$

• Ohm's law and the power equation work if one uses the RMS quantities of everything.

Capacitors in AC Circuits

• Reactance (symbolized by $X$) is the equivalent to resistance as frequency can adjust the relationship between current and voltage. The opposition to AC current is inversely proportional to frequency

• $X_C = \frac{1}{\omega C}$

  • $X_C$ is the capacitive reactance.

  • $\omega$ is the angular frequency.

  • $C$ is the capacitance.

• Capacitor current leads the voltage by 90°.

  This is because when the voltage is first truned on the capacitor acts like a wire, very quickly starting to discharge. Once the max volatge is reached, then this current starts to decrease. This repeates in the other direction.

Impedance

• The impedance vector in capacitors is what resistance is in standard current.

*Impedence is still a measurement of voltage, but frequency also influences this value.

• The total resistance in an AC circuit is called the impedance.

• Represented by "Z".

Magnets and the Biot-Savart Law

• Metals have magnetic poles- North and South
  *opposite charges attract, same charges repel.

• Moving charges create magnets.

• Biot-Savart law is the equation to measure the magnetic field strength:

  $dB = (\frac{\mu_0}{4\pi}) \frac{Idl \times \hat r}{r^2}$

  Biot-Savart simplified:

  *point index finger in direction of positive charge and middle finger from wire to observation point, thumb directs magnetic field.

  • Current = direction of positive charges

  • magnetic field proportional to current

  • magnetic field inversely proportional to r^2

Magnetic Field around a wire

*Thumb in direction of the current, direction of fingers curl is the direction the magnetic field is pointing.

*With many loops, can make consistent mgnetic field.

Lorentz Law and Forces

*Equation for calculating the force of a point of charge due to magnetic field: $\vec F = q \vec v \times \vec B$

• If no velocity, no magnetic force.
  *perpendicular velocity == perpendicular mgnetic field
  *Otherwise, it experiences both that are perpendicular to both velocity and mgnetic field. (see cross product right hand rule notes)

• Charges will follow a helical path when going through a magnetic field.

Measuring Magnetic Field (Hall Effect)

1. Position the magnetic field perpendicular to the current.

2. Charges move to make electric feild with electric feild force to negate magnetic feild. Continue to move charges till balance.

3. After balance, $"Q * E = Q * v cross B$

4. Solve for B in equation (B = \frac{V}{vd})

• B = Magnetic Field, Q= charge, v= velocity, V=Voltage
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