Test Compilation PHY 102 15

This study summary covers key concepts in electricity and magnetism, including circuit analysis, electromagnetism, solid-state physics, and electrostatics, as derived from the provided questions. ---

Electromagnetism & Circuits

Induced Electromotive Force (EMF)

• Faraday's Law of Induction: The magnitude of induced e.m.f is directly proportional to the rate of change of magnetic flux linkage.
• Induced EMF in a Moving Conductor: When a conductor moves through a magnetic field, the induced voltage (EMF) increases with the speed of the conductor (EMF = Blvsinθ).
• Induced EMF in a Straight Wire: For a straight wire of length `L` moving at speed `v` at an angle `θ` to a magnetic field `B`, the induced e.m.f is given by `E = BLvsinθ`.

Magnetic Fields & Forces

• Faraday's Law (Conceptual): A changing magnetic field produces an induced electric field.
• Magnetic Field at Center of a Coil: For a circular coil of `N` turns and radius `R` carrying current `I`, the magnetic field at its center is `B = (μ₀NI) / (2R)`. If a wire is initially bent into one turn (radius `R`) with field `B`, and then bent into `n` turns, the new radius `R_n = R/n`. The new magnetic field `B_n = (μ₀nI) / (2R_n) = (μ₀nI) / (2R/n) = n²B`. Therefore, the magnetic field will be proportional to n²B.
• Torque on a Current Loop: The torque `τ` on a loop of `N` turns, area `A`, carrying current `I` in a magnetic field `B` is `τ = NIABsinθ`. If the plane of the loop is parallel to the magnetic field, `θ = 90°` and `sinθ = 1`, so `τ = NIAB`.
• Electron in Magnetic Field: An electron moving perpendicular to a uniform magnetic field follows a circular path. The radius `r` of this path is given by `r = mv / (qB)`, where `m` is electron mass, `v` is speed, `q` is electron charge, and `B` is magnetic field density.
• Magnetic Flux: The magnetic flux `Φ` through a coil of area `A` in a magnetic field `B` is `Φ = BAcosθ`, where `θ` is the angle between the magnetic field and the normal to the area. If the plane is inclined at 60° to the field, the angle with the normal is 30°, so `Φ = BAcos(30°)`.

Circuit Components & Laws

• Ohm's Law: Applicable only to linear circuits or components.
• Kirchhoff's Current Law (Junction Rule): The sum of currents entering a junction equals the sum of currents leaving it.
• Resistors in Series/Parallel: Understand how to calculate total resistance for given configurations. (Example: A bridge circuit or series-parallel combinations).
• Current in a Conductor (No Potential Difference): Current is zero because for every electron with a given velocity, there is another with a velocity of equal magnitude and opposite direction, leading to zero net flow.
• S.I. Unit of Potential Difference: The S.I. unit of potential difference is the Volt (V), which is equivalent to Joules per Coulomb (J/C).
• Maximum Power Transfer Theorem: Maximum power is delivered from a source to a variable external resistor `R` when the external resistance `R` is equal to the internal resistance `r` of the source (`R=r`).
• EMF vs Terminal Potential Difference: The EMF of a battery is equal to its terminal potential difference only when no current is drawn (open circuit) or when its internal resistance is negligible.

Capacitors

• Capacitors in Series: When three capacitors are joined in series, the total capacitance is less than the value of the minimum capacitance (1/C_total = 1/C₁ + 1/C₂ + 1/C₃).
• Capacitor Current-Voltage Relation: The current `I` through a capacitor is proportional to the rate of change of voltage across it: `I = C dV/dt`.
• DC Voltage on a Capacitor: When DC voltage is applied to a capacitor, it acts as a short circuit at the instant of applied voltage and then as an open circuit once fully charged.
• Parallel Plate Air Capacitor (Charge Calculation): `Q = CV`, where `C = ε₀A/d`. (where `ε₀` is permittivity of free space, `A` is plate area, `d` is plate separation).
• Dielectric Insertion (Battery Connected): If a dielectric slab is inserted into a parallel plate capacitor while connected to a battery, the capacitance and charge on the positive plate increase, but the potential difference between the plates remains the same (because the battery maintains constant voltage).
• Parallel Plate Capacitor (Disconnected, Plates Pulled Apart): If a charged parallel-plate capacitor is disconnected from the battery and the plates are pulled apart to twice their original separation, the stored energy will double (since `Q` remains constant, `C` halves, and `V` doubles, `E = Q²/2C = ½QV`). The electric field between the plates remains constant. The ratio of capacitance before and after inserting a metal sheet of thickness d/2 is typically 1:1 if the metal sheet effectively divides the capacitor into two in series, but with the given options and specific context (often simplified), the ratio of C_before:C_after insertion of d/2 metal sheet is 1:2.

AC Circuits (RLC and Inductors)

• R-L-C Circuit (Maximum Current Flow): In a series R-L-C circuit, maximum current flow occurs at resonance, where inductive reactance (`X_L`) equals capacitive reactance (`X_C`). The resonant frequency is `f = 1 / (2π√(LC))`.
• RL Circuit Time Constant: For an RL circuit, the time constant `τ = L/R`. Doubling the inductance `L` will increase the time constant to twice its original value.
• Apparent Power (RL Circuit): In an AC RL circuit, apparent power `S = V_rms * I_rms`. The impedance `Z = √(R² + X_L²)`, where `X_L = 2πfL`. So `S = V_rms² / Z`.
• Impedance (RL Circuit, X_L=R): If the inductive reactance `X_L` is equal to the resistance `R`, the impedance `Z = √(R² + X_L²) = √(R² + R²) = R√2`.

Mutual Inductance

• Mutual Inductance (M): For a transformer, the induced EMF in the secondary coil `e₂` is related to the rate of change of current in the primary coil `dI₁/dt` by `e₂ = -M (dI₁/dt)`. Thus, `M = -e₂ / (dI₁/dt)`.

Capacitor Charging (RC Circuit)

• Voltage Across Resistor during Charging: In an RC circuit, the voltage across the resistor during charging is `V_R(t) = V_₀ * e^(-t/RC)`. (where `V_₀` is battery voltage, `R` is resistance, `C` is capacitance).

Electrostatics

Coulomb's Law

• Assumptions: Coulomb's law applies to bodies that are small compared to the distance between them.
• Force Relationship: The electrostatic force `F` (attraction or repulsion) is directly proportional to the product of the two charges and inversely proportional to the square of the distance between them. (`F = k * q₁q₂ / r²`).
• Number of Excess Electrons: To find the number of electrons `n` for a given force, use `q = ne`, then `F = k * (ne)² / r²`.
• Coulomb's Constant (k): The constant `k` in Coulomb's law is dimensionally equivalent to `[ML³T⁻⁴I⁻²]`.

Electric Fields & Potential

• Electric Field Direction: The electric field `E` at a point P is radially outward if the source charge is positive and radially inward if the source charge is negative.
• Electric Field from Potential: If the potential `V(x,y)` is given, the electric field components are `E_x = -∂V/∂x` and `E_y = -∂V/∂y`. The magnitude is `|E| = √(E_x² + E_y²)`.
• Gauss's Law Applications:
  • For spherical charge distribution, the Gaussian surface is a sphere.
  • For a charge distribution with the symmetry of an infinite uniform plane, the Gaussian surface is a "coin-shaped" flat cylinder or rectangular box.
  • For an infinitely long cylinder or wire, the Gaussian surface is a coaxial cylinder.
• Electric Field Outside Spherical Gaussian Surface: For a point charge or spherically symmetric charge distribution, `E = kQ/r²`, where `r` is the distance from the center. (Careful with diameter vs. radius and distance from origin).
• Electric Field of a Hollow Metallic Sphere: The electric field associated with a uniformly charged hollow metallic sphere is greatest at its surface. Inside it is zero, and outside it decreases with `1/r²`.

Charge Transfer

• Rubbing Fur on Rubber Rod: When fur rubs a hard rubber rod, electrons are transferred from the fur to the rod, making the rod negatively charged. The fur loses electrons and becomes positively charged (not neutral).

Solid State Physics

Semiconductors

• Resistance & Temperature: The resistance of semiconductors decreases with increasing temperature.
• Doping: The introduction of small amounts of impurity into pure semiconductors is called doping.
• N-type vs P-type Semiconductors: Doping silicon with boron (a trivalent impurity) creates P-type semiconductors by introducing "holes" (electron acceptors), not N-type.
• Strength of Crystal: The strength of a semiconductor crystal comes from electron-pair bonds (covalent bonds).

Energy Bands in Solids

• Conduction Band: An electron in the conduction band has higher energy than an electron in the valence band.
• Forbidden Bands: The part of a solid which electrons cannot occupy is called the forbidden band (or energy gap).

Diodes

• Diode Conduction: Under normal conditions, a diode conducts current when it is forward biased.

Measurement & Devices

Galvanometer

• Galvanometer Sensitivity: The sensitivity of a galvanometer is given by `S = (NBA) / k`, where `N` is the number of turns, `B` is the magnetic field strength, `A` is the coil area, and `k` is the spring constant.
• Ammeter Reading: An ammeter measures current. If a magnet is sitting motionless next to a coil, there is no change in magnetic flux, hence no induced EMF or current according to Faraday's law. Therefore, the ammeter should indicate that there is no current flowing (i.e., read zero).

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