Exhaustive University Study Notes: Electromagnetism and AC Circuit Theory and Atomic Physics

AC Circuits, Resistance, and Resonance

• Capacitive Reactance ($X_C$): For an $8.00 \, \mu F$ capacitor connected to a $60.0 \, Hz$ AC source, the capacitive reactance is calculated as $X_C = \frac{1}{2\pi f C}$, resulting in approximately $332 \, \Omega$ (specifically $331.572 \, \Omega$).
• Series RLC Impedance ($Z$): In a circuit with $R = 425 \, \Omega$, $L = 1.25 \, H$, and $C = 3.50 \, \mu F$ at an angular frequency $\omega = 377 \, s^{-1}$ and $\Delta V_{max} = 150 \, V$, the impedance is approximately $513 \, \Omega$. In a separate example with $R = 200 \, \Omega$, $X_L = 40 \, \Omega$, and $X_C = 1000 \, \Omega$, the impedance is $Z = 980 \, \Omega$.
• Resonance Properties:
  • Resonance occurs when inductive reactance equals capacitive reactance ($X_L = X_C$).
  • At resonance, the impedance of a series RLC circuit is equal to the resistance ($R$).
  • The current in a series RLC circuit reaches its maximum value at the resonance frequency.
  • Increasing the resistance of a coil does not increase the $Q$ (Quality Factor) of the coil at resonance (False).
• Quality Factor ($Q$) and Bandwidth:
  • The Quality Factor is a dimensionless quantity (SI unit: No unit).
  • An increase in bandwidth results in a decrease in the Quality Factor.
  • It is false to say that a higher Quality Factor leads to a greater operating bandwidth; higher $Q$ actually implies a narrower, sharper resonance (False).
• Power and Phase in AC Circuits:
  • RMS (Root Mean Square): Represents the effective value of AC current or voltage.
  • Power Factor: For a circuit with $R = 15 \, \Omega$, $L = 25 \, mH$, and $C = 35 \, \mu F$ at $100 \, Hz$, the power factor is $0.45$.
  • In a purely inductive circuit, the current lags the voltage by one-fourth of a cycle ($90^{\circ}$, or $\frac{\pi}{2}$).
  • In a purely capacitive circuit, the current leads the voltage by one-fourth of a cycle ($90^{\circ}$, or $\frac{\pi}{2}$).
  • In a purely resistive circuit, the current and voltage are in phase.
  • Average power at resonance (given $V_{max} = 100 \, V$ and $R = 100 \, \Omega$) is approximately $49.98 \, W$.
  • The average value of a sinusoidal AC current over one full cycle is zero.
• Voltage and Current Calculations:
  • For a source $\Delta V = 200 \sin(\omega t)$ connected to a $100 \, \Omega$ resistor, the RMS current ($I_{rms}$) is $1.41 \, A$.
  • For a lightbulb consuming an average power of $75.0 \, W$ with a $\Delta V_{max} = 170 \, V$, the resistance is $193 \, \Omega$.

Fundamental Properties of Magnetic Fields and Forces

• Sources of Magnetic Fields: The region of space surrounding any moving electric charge contains a magnetic field. However, constant current in a helical coil causes the coil to tend to get shorter due to magnetic attraction between adjacent turns.
• Magnetic Poles:
  • Magnetic poles are always found in pairs (dipoles); a single magnetic pole (monopole) has never been isolated.
  • Every magnet has two poles regardless of shape. Dividing a bar magnet in two creates two new, complete bar magnets.
  • The Earth's magnetic north pole is not the same as the geographic north pole (False).
• Magnetic Force on Moving Charges:
  • The magnetic force acts on a charged particle only when it is in motion.
  • The magnitude of the force is proportional to $\sin(\theta)$, where $\theta$ is the angle between the velocity vector and the magnetic field ($B$).
  • Force is zero if the particle moves parallel or anti-parallel to the magnetic field vector ($\theta = 0^{\circ}$ or $180^{\circ}$).
  • An electron traveling due north in a magnetic field directed due north will be unaffected by the field.
  • Kinetic Energy: The magnetic field alone cannot alter the kinetic energy of a charged particle because the magnetic force is always perpendicular to the velocity (doing zero work).
• Magnetic Force on Current-Carrying Wires:
  • A wire suspended between magnetic poles experiences a deflection that depends on the direction of both the field and the current.
  • The force is maximum when the wire is perpendicular to the magnetic field and minimum (zero) when the wire is parallel to the field.
  • Lorentz Force: The combined force on a charge in both electric ($E$) and magnetic ($B$) fields is given by $\mathbf{F} = q\mathbf{E} + q\mathbf{v} \times \mathbf{B}$ (The transcript notes the specific expression $F = qB + qvE$ is False).

Motion in Magnetic Fields and Applications

• Circular Motion:
  • When velocity is perpendicular to a uniform magnetic field, the particle follows a circular path. The radius of this path is given by $r = \frac{mv}{qB}$.
  • For a proton ($m_p = 1.67 \times 10^{-27} \, kg$) in a $0.5 \, T$ field with a radius of $10 \, cm$, the speed is $4.8 \times 10^{6} \, m/s$.
  • The angular speed for a proton in a $1.67 \, T$ field with radius $20 \, cm$ is $1.6 \times 10^{8} \, rad/s$.
• Velocity Selector:
  • In a velocity selector, particles move in a straight line when the electric and magnetic forces balance, requiring a velocity $v = \frac{E}{B}$.
  • A filter with $E = 15 \times 10^{6} \, V/m$ and $B = 1.5 \times 10^{2} \, T$ selects for a velocity of $10^{5} \, m/s$.
• Mass Spectrometer: This device separates ions based on their mass-to-charge ratio ($m/q$).
• Torque on Current Loops:
  • Torque is maximum when the magnetic field is parallel to the plane of the loop (or perpendicular to the normal of the loop).
  • The magnetic dipole moment of a loop is $\mu = IA$.
  • For a rectangular coil ($5 \, cm \times 8 \, cm$, $100$ turns, $10 \, mA$, $0.5 \, T$) where the field is parallel to the plane, the torque is $2 \times 10^{-3} \, N \cdot m$ and the dipole moment is $4 \times 10^{-3} \, A \cdot m^2$.
• Van Allen Radiation Belts: These consist of charged particles trapped by Earth's magnetic field. Auroras are confined to polar regions because these belts are nearest the Earth's surface at the poles.

Electrostatics: Fields, Forces, and Flux

• Coulomb’s Law and Electric Forces:
  • Electric field lines for a positive charge are directed radially outward; for a negative charge, they are radially inward.
  • Charges of the same sign repel; charges of opposite signs attract.
  • The electric force is inversely proportional to the square of the separation ($r^2$). If distance triples, force decreases to $\frac{1}{9}$ of the original value.
  • Field strength at a point is measured by the force experienced by a unit test charge ($E = \frac{F}{q}$).
• Charge Density:
  • Linear Charge Density ($\lambda$): Charge per unit length ($C \cdot m^{-1}$).
  • Surface Charge Density ($\sigma$): Charge per unit area ($C \cdot m^{-2}$). If $1 \, C$ is on a surface of $1 \, cm^2$, $\sigma = 1 \, C/cm^2$.
  • Volume Charge Density ($\rho$): Charge per unit volume ($C \cdot m^{-3}$).
• Electric Field Calculations:
  • The field due to a point charge is $E = \frac{kQ}{r^2}$. If the charge is doubled at the same distance, the field becomes $2E$. If the distance doubles, the field becomes $\frac{E}{4}$.
  • Field inside a thin spherical shell is zero. Field outside follows the point charge formula.
  • Field inside a uniformly charged insulating solid sphere is proportional to the distance from the center ($r$).
• Gauss’s Law and Electric Flux ($\Phi_E$):
  • Gauss’s Law relates total flux through any closed surface to the net enclosed charge: $\Phi_E = \frac{Q_{enclosed}}{\epsilon_0}$.
  • Flux through a closed surface is independent of the shape of the surface (e.g., changing from a sphere to a cube does not change the flux).
  • The net electric flux through a closed surface that surrounds no charge is zero.
  • Flux is maximum when field lines are perpendicular to the area surface (parallel to the normal vector).
  • Flux Scenario: With charges $-2Q, Q$, and $-Q$, the flux through surface $S_3$ is $-\frac{2Q}{\epsilon_0}$; the flux through $S_4$ is zero.

Magnetic Induction and Solenoids

• Faraday’s Law: States that induced EMF is proportional to the rate of change of magnetic flux.
• Solenoids and Toroids:
  • The magnetic field inside a long solenoid is $B = \mu_0 n I$, where $n$ is the number of turns per unit length.
  • Doubling the radius of a solenoid does not change the interior magnetic field magnitude.
  • Doubling both the number of turns and the length results in no change to the magnetic field (as $n$ remains constant).
  • For a toroid ($r_{inner} = 0.8 \, m, r_{outer} = 1.5 \, m, 1000 \, turns, 15 \, kA$), the field at the inner radius is $3.75 \, T$.
• Induction: An EMF is induced in a wire by moving it relative to a magnet. A changing magnetic field can produce an electric current.
• Units:
  • Magnetic Field Units: Tesla ($T$), where $1 \, T = 10^4 \, Gauss$. Alternatively, $kg \cdot C^{-1} \cdot s^{-1}$ or $N \cdot A^{-1} \cdot m^{-1}$.
  • Permeability of Free Space ($\mu_0$): Units are $T \cdot m \cdot A^{-1}$.
  • Magnetic Flux Unit: Weber ($Wb$), where $1 \, Wb = 1 \, T \cdot m^2$.

Questions & Discussion

• Q: What happens if you reverse the direction of current in a wire?   A: The direction of the resulting magnetic field is reversed.
• Q: Does magnetic field point north to south?   A: Outside a magnet, field lines point from the North pole to the South pole. (The statement that they point South to North is False).
• Q: Is the electric field vector?   A: Yes, the electric field is a vector quantity (The statement that it is scalar is False).
• Q: Can electric field lines cross?   A: No, field lines can never cross each other.
• Q: What is the force between two parallel wires?   A: Wires carrying current in the same direction attract; opposite directions repel.
• Q: What is the speed of an electron released in a $520 \, N/C$ field after $48 \, ns$?   A: Estimations show a speed of approximately $4.39 \times 10^6 \, m/s$.