Test Compilation PHY 102 17

This study summary covers various topics in physics based on the provided questions, including Electricity and Magnetism, AC Circuits, DC Circuits, Semiconductors, and Electrostatics.

Electricity and Magnetism

• Inductance:
  • Unit: Henry (H)
• Moving Coil Galvanometer:
  • Concave pole faces are used to produce a radial magnetic field and increase the field strength, ensuring a linear scale for current measurement.
• Cyclotron:
  • Used to accelerate charged particles.
• Force between Two Parallel Current-Carrying Wires:
  • Formula: $F/L = (\mu_0 I_1 I_2) / (2\pi d)$
  • Where:
    • $F/L$ = Force per unit length ($N/m$)
    • $\mu_0$ = Permeability of free space ($4\pi \times 10^{-7} \text{ Tm/A}$)
    • $I_1, I_2$ = Currents in the wires ($A$)
    • $d$ = Distance between wires ($m$)
  • Example: For $I_1=20A$, $I_2=30A$, $d=40mm$, $F/L = 0.003 \text{ N/m}$ (attractive if currents are in the same direction).
• Torque on a Current Loop in a Magnetic Field:
  • Formula: $\tau = NIAB \sin\theta$
  • Maximum torque occurs when $\sin\theta = 1$, so $\tau_{max} = NIAB$.
  • Where:
    • $\tau$ = Torque ($Nm$)
    • $N$ = Number of turns in the coil
    • $I$ = Current ($A$)
    • $A$ = Area of the coil ($m^2$)
    • $B$ = Magnetic field strength ($T$)
  • Example: For $N=45$, $A=1.8 \times 10^{-2} \text{ m}^2$, $\tau_{max}=2.5 \times 10^{-2} \text{ Nm}$, $B=0.08 \text{ T}$, the current $I \approx 0.39 \text{ A}$.
• Magnetic Field Lines Characteristics:
  • They form closed loops.
  • They emerge from the North pole and enter the South pole (outside the magnet).
  • They do not intersect each other.
  • The density of lines indicates the strength of the magnetic field.
  • The tangent to a field line at any point gives the direction of the magnetic field at that point.
• Magnetic Flux:
  • Formula: $\Phi_B = \vec{B} \cdot \vec{A} = BA \cos\theta$
  • Where $\theta$ is the angle between the magnetic field vector ($\vec{B}$) and the normal to the surface area ($\vec{A}$).
  • If the magnetic field is parallel to the plane of the area, then $\theta = 90^\circ$, and $\Phi_B = 0$.
• Induced EMF and Faraday's Law:
  • The phenomenon of producing an induced emf and current due to a change in magnetic flux is called electromagnetic induction.
  • Induced EMF is maximum when the rate of change of magnetic flux linkage is maximum. This occurs when the magnetic flux itself is momentarily zero (e.g., when the plane of a rotating coil is parallel to the magnetic field).
• Force on a Current-Carrying Wire in a Magnetic Field:
  • Formula: $F = BIL \sin\theta$
  • For a wire placed perpendicular to the magnetic field ($\theta = 90^\circ$), $F = BIL$.
  • Where:
    • $F$ = Force ($N$)
    • $B$ = Magnetic field strength ($T$)
    • $I$ = Current ($A$)
    • $L$ = Length of the wire ($m$)
  • Example: For $L=0.05m$, $I=30A$, $B=0.8T$, the force $F = 1.2 \text{ N}$.
• Force on a Charged Particle in a Uniform Magnetic Field:
  • Formula: $\vec{F} = q(\vec{v} \times \vec{B})$
  • The magnetic force is always perpendicular to the velocity of the particle.
  • This force does no work on the particle; therefore, it changes the direction of the velocity but not its magnitude (speed) or kinetic energy. A charged particle released in a uniform magnetic field (with no other forces) will exhibit constant speed.

Electrostatics

• Acceleration of a Positively Charged Particle in an Electric Field:
  • Formula: $\vec{F} = q\vec{E}$. Since $\vec{F} = m\vec{a}$, then $\vec{a} = (q/m)\vec{E}$.
  • For a positive charge ($q > 0$), the acceleration is in the same direction as the electric field.
• Coulomb's Law:
  • Formula: $F = k |q_1 q_2| / r^2$
  • Where:
    • $F$ = Force ($N$)
    • $k$ = Coulomb's constant ($9 \times 10^9 \text{ N m}^2/\text{C}^2$)
    • $q_1, q_2$ = Magnitudes of charges ($C$)
    • $r$ = Distance between charges ($m$)
  • Repulsive Force: Occurs between similar charges (both positive or both negative).
  • Attractive Force: Occurs between opposite charges (one positive, one negative).
  • Example: For $q_1 = -2.3 \times 10^{-6} \text{ C}$, $F=0.35 \text{ N}$ (repulsive), $r=0.20 \text{ m}$, the unknown charge $q_2 = -6.76 \times 10^{-7} \text{ C}$ (negative polarity, magnitude $6.8 \times 10^{-7} \text{ C}$).
  • Graphical Analysis: Plotting $\log F$ vs $\log r$ for Coulomb's law gives a straight line with a slope of -2 (since $F \propto r^{-2}$).
• Electric Flux through a Cylindrical Surface in a Uniform Electric Field:
  • For a uniform electric field $E$ and a cylindrical surface of radius $r$ and height $h$, if the "maximum flux" is taken to be the flux through the curved surface when the field is idealized to be perpendicular to it everywhere (or through the end caps when field is parallel to axis), the value can be $E \times (2\pi r h)$.
  • Example: For $r = 0.05 \text{ m}$, $h = 0.025 \text{ m}$, $E = 250 \text{ N/C}$, the flux $\Phi_E = 250 \times (2\pi \times 0.05 \times 0.025) \approx 1.963 \text{ Nm}^2/\text{C}$.
• Gauss's Law:
  • States that the total electric flux ($\Phi_E$) through any closed surface (Gaussian surface) is proportional to the total electric charge ($Q_{enc}$) enclosed within that surface: $\Phi_E = Q_{enc} / \epsilon_0$.
  • The net charge enclosed is the algebraic sum of positive and negative charges and can be positive, negative, or zero. It is a scalar, not a vector.
  • The total flux emerging from a given charge is fundamentally independent of the medium surrounding it in the sense that the proportionality constant $\epsilon_0$ is for vacuum. If the Gaussian surface is within a dielectric, the permittivity $\epsilon$ would be used instead of $\epsilon_0$.
  • Gauss's Law is primarily useful for charge distributions with high symmetry for calculating the electric field:
    • For spherical charge distribution, use a spherical Gaussian surface.
    • For an infinitely long line of charge or cylinder, use a coaxial cylindrical Gaussian surface.
    • For an infinite plane of charge, use a cylindrical (coin-shaped) Gaussian surface with ends parallel to the plane.
  • The statement "The net charge is in the direction of the applied field" is incorrect. Charge is a scalar.
• Electric Potential Energy and Work Done:
  • Work done by an external agent to move a charge $q_0$ from point X to point Y in an electric field is $W = q_0 (V_Y - V_X)$.
  • Example: For $q_0 = 20 \mu C$ and $V_Y - V_X = 35 \text{ V}$, the work done is $W = (20 \times 10^{-6} \text{ C}) \times (35 \text{ V}) = 7 \times 10^{-4} \text{ J}$.
  • To convert Joules to electron-volts (eV): $1 \text{ eV} = 1.602 \times 10^{-19} \text{ J}$. So $7 \times 10^{-4} \text{ J} \approx 4.37 \times 10^{15} \text{ eV}$.
• Units of Electric Intensity (Electric Field Strength):
  • Standard units are Newton per Coulomb ($N/C$) or Volt per meter ($V/m$).
  • Joule per Coulomb-meter ($J/Cm$) is also equivalent to $N/C$, as $J = Nm$, so $J/Cm = (Nm)/(Cm) = N/C$.
  • Joule per Coulomb ($J/C$) is the unit for electric potential (Volt), not electric field intensity. So, $J/C$ cannot be a unit of electric intensity.

Capacitance and Dielectrics

• Capacitance of a Parallel-Plate Capacitor:
  • Formula: $C = \epsilon_0 A / d$ (in vacuum/air)
  • Where:
    • $C$ = Capacitance ($F$)
    • $\epsilon_0$ = Permittivity of free space ($8.854 \times 10^{-12} \text{ F/m}$)
    • $A$ = Area of plates ($m^2$)
    • $d$ = Distance between plates ($m$)
  • Example 1: For $A=4 \text{ cm}^2$ and $d=1 \text{ mm}$, $C \approx 3.54 \text{ pF}$.
  • Example 2: For $A=15 \text{ cm}^2$ and $d=3 \text{ cm}$, $C \approx 4.43 \times 10^{-13} \text{ F}$.
  • If the distance between plates ($d$) is halved, the capacitance ($C$) doubles.
• Capacitors with Dielectrics:
  • When a dielectric material with dielectric constant $K$ is inserted between the plates, the capacitance increases: $C = K C_0$.
  • Example: If $C_0 = 20 \mu F$ and $K=2$, then $C = 40 \mu F$.
  • Dielectrics are non-conducting materials that, when placed between capacitor plates, increase the capacitance and decrease the electric field inside the capacitor (for a given charge).
• Induced Charge in a Dielectric:
  • If a capacitor is charged to $V_0$ and then disconnected, and a dielectric (K) is inserted, the voltage decreases to $V = V_0/K$. The initial charge $Q_0$ remains constant.
  • The induced charge on the dielectric surfaces is $Q_{ind} = Q_0 (1 - 1/K)$.
  • Example: For $A=0.2 \text{ m}^2$, $d=0.01 \text{ m}$, $V_0=3000 \text{ V}$, and $V=1000 \text{ V}$ (implying $K=3$), $Q_{ind} \approx 3.54 \times 10^{-7} \text{ C} = 35.4 \times 10^{-8} \text{ C}$.
• Relation between Current and Voltage in a Capacitor:
  • The capacitance is defined as $C = Q/V$.
  • The current is the rate of change of charge: $I = dQ/dt$.
  • From $Q=CV$, differentiating with respect to time gives: $I = C (dV/dt)$.
• Energy Stored in a Capacitor:
  • Formula: $U_E = (1/2)CV^2 = (1/2)QV = Q^2/(2C)$
  • Where:
    • $U_E$ = Stored Energy ($J$)
    • $C$ = Capacitance ($F$)
    • $V$ = Voltage ($V$)
    • $Q$ = Charge ($C$)
  • Work done in charging a capacitor is equal to the stored energy, commonly expressed as $(1/2)QV$.
  • Example: For $C=2000 \text{ mF} = 2 \text{ F}$ and $V=10 \text{ V}$, $U_E = (1/2) \times 2 \times (10)^2 = 100 \text{ J}$.
• RC Circuits (Capacitor Charging):
  • If a capacitor $C$ is charged through a resistor $R$ by a voltage source $V_S$:
    • Maximum charging current ($t=0$): $I_{max} = V_S / R$.
    • Charge stored when fully charged ($t \to \infty$): $Q_{max} = C V_S$.
  • Example: If $C=4.0 \mu F$, $R=2.5 \text{ M}\Omega$, and assuming $V_S=10 \text{ V}$, then $I_{max}=4 \mu A$ and $Q_{max}=40 \mu C$.

DC Circuits

• Equivalent Resistance:
  • Series: $R_{eq} = R_1 + R_2 + ...$
  • Parallel: $1/R_{eq} = 1/R_1 + 1/R_2 + ...$
  • Example 1: Four 20-ohm resistors in parallel: $R_{eq} = 20/4 = 5 \Omega$. Current for 20V emf: $I = V/R_{eq} = 20V / 5\Omega = 4A$.
  • Example 2: Three 20-ohm resistors in parallel: $R_{eq} = 20/3 \approx 6.67 \Omega$.
  • Example 3: A 20k$\Omega$ and 40k$\Omega$ resistor in series ($R_{series} = 60\text{k}\Omega$). If an unknown resistor $R$ is connected in parallel with $R_{series}$ to give a total equivalent resistance of 20k$\Omega$, then $R = 30\text{k}\Omega$.
• Ohm's Law:
  • $V = IR$
• Resistance of a Conductor:
  • Formula: $R = \rho L / A$
  • Where:
    • $R$ = Resistance ($\Omega$)
    • $\rho$ = Resistivity of the material ($\Omega \cdot m$)
    • $L$ = Length of the conductor ($m$)
    • $A$ = Cross-sectional area ($m^2$)
  • Resistivity of metals generally increases with temperature.
  • To minimize resistance, choose a conductor that is thick, short, and cool.
• Battery Terminal Voltage with Internal Resistance:
  • Formula: $V_T = \mathcal{E} - Ir$
  • Current in circuit: $I = \mathcal{E} / (R_{ext} + r)$
  • Where:
    • $V_T$ = Terminal voltage ($V$)
    • $\mathcal{E}$ = EMF of the battery ($V$)
    • $I$ = Current ($A$)
    • $r$ = Internal resistance of the battery ($\Omega$)
    • $R_{ext}$ = External resistance ($\Omega$)
  • Example: For $\mathcal{E}=4V$, $r=2\Omega$, $R_{ext}=8\Omega$, then $I = 0.4A$ and $V_T = 3.2V$.
• Kirchhoff's Current Law (KCL):
  • KCL states that the algebraic sum of currents entering a node (or junction) in an electrical circuit is equal to zero.
  • This law is based on the conservation of electric charge.
• Capacitors in Series:
  • Equivalent capacitance: $1/C_{eq} = 1/C_1 + 1/C_2 + ...$
  • The charge on each capacitor in a series connection is the same and equal to the total charge stored by the equivalent capacitance: $Q_{total} = C_{eq} V_{source}$.
  • Example: For $C_1=6mF$ and $C_2=3mF$ in series across $18V$, $C_{eq}=2mF$, and $Q_{total}=36mC$. Thus, each capacitor has $36mC$ of charge.

AC Circuits

• Sinusoidal AC Waveform:
  • General Voltage Equation: $v = V_{peak} \sin(\omega t)$
  • Angular Frequency: $\omega = 2\pi f = 2\pi / T$
  • RMS Value: $V_{rms} = V_{peak} / \sqrt{2}$; $I_{rms} = I_{peak} / \sqrt{2}$
  • Example 1: For a period $T = 0.005 \text{ s}$, $\omega = 2\pi / 0.005 = 400\pi \approx 1257 \text{ rad/s}$. So the waveform equation is $v = V \sin(1257t)$.
  • Example 2: For a current $I = 15 \sin(\omega t)$, $I_{peak}=15A$. The RMS current $I_{rms} = 15 / \sqrt{2} \approx 10.6 \text{ A}$.
  • Example 3: For $v = 290 \sin(1256 t)$, $V_{peak}=290V$. $V_{rms} = 290/\sqrt{2} \approx 205 V$. Period $T = 2\pi/1256 \approx 0.005 \text{ s} = 5 \text{ ms}$.
• RLC Series Circuit:
  • Inductive Reactance: $X_L = \omega L$
  • Capacitive Reactance: $X_C = 1 / (\omega C)$
  • Impedance: $Z = \sqrt{R^2 + (X_L - X_C)^2}$
  • RMS Current: $I_{rms} = V_{rms} / Z$
  • Example: For $R=100\Omega$, $L=1H$, $C=10\mu F$, $\omega=400 \text{ rad/s}$, and $V_{peak}=200V$:
    • $X_L = 400\Omega$
    • $X_C = 250\Omega$
    • $Z \approx 180.28\Omega$
    • $V_{rms} \approx 141.42V$
    • $I_{rms} \approx 0.78 \text{ A}$.
• Capacitor in AC Circuit (Phase Relation):
  • In a purely capacitive AC circuit, the current leads the voltage by $90^\circ$ (or the voltage lags the current by $90^\circ$).
• Transformers and Efficiency:
  • Efficiency: $\eta = (P_{out} / P_{in}) \times 100\%$
  • Output Power: $P_{out} = V_{out} I_{out}$
  • Example: A transformer with $80\%$ efficiency has an output of $12\text{V}$ and $4\text{A}$.
    • $P_{out} = 12\text{V} \times 4\text{A} = 48\text{W}$.
    • $P_{in} = P_{out} / \eta = 48\text{W} / 0.8 = 60\text{W}$.

Semiconductors

• Energy Bands:
  • The energy region which electrons cannot occupy in a solid is called the forbidden energy gap.
  • Free electrons exist in the conduction band.
• Doping:
  • Doping is the process of adding impurities to a semiconductor material to alter its electrical conductivity.
  • Doping a quadrivalent element (like Silicon) with a pentavalent element (like Phosphorus) results in an N-type semiconductor. This creates donor energy levels just below the conduction band, increasing the electron concentration.
  • The resistivity of a semiconductor depends significantly on its temperature, doping concentration, and intrinsic material properties.
• Conduction in Semiconductors:
  • Conduction in semiconductors occurs due to the movement of both electrons (in the conduction band) and holes (in the valence band).
  • Intrinsic conduction: Conduction in an undoped semiconductor, where electron and hole concentrations are equal. There are no majority or minority carriers.
  • Extrinsic conduction: Conduction in a doped semiconductor, where conduction is dominated by majority carriers (electrons in N-type, holes in P-type).
• PN Junctions (Forward Bias):
  • For a PN junction to be forward biased, the positive terminal of the battery must be connected to the P-side, and the negative terminal to the N-side.

Mechanics and Efficiency

• Motor Efficiency:
  • Efficiency $\eta = (P_{out} / P_{in}) \times 100\%$
  • Electrical Input Power: $P_{in} = V I$
  • Mechanical Output Power (for lifting a load): $P_{out} = F v = (mg) v$
  • Example: A motor (240V, 12A) lifts a 600kg load at 0.15m/s.
    • $P_{in} = 240\text{V} \times 12\text{A} = 2880\text{W}$.
    • $P_{out} = (600\text{kg} \times 9.8\text{m/s}^2) \times 0.15\text{m/s} = 882\text{W}$.
    • $\eta = (882/2880) \times 100\% \approx 30.6\%$.

Other Concepts

• Triboelectric Effect (Glass Rod and Silk):
  • When a glass rod is rubbed with silk, the glass rod loses electrons and becomes positively charged, while the silk gains electrons and becomes negatively charged.