Physics

Mechanics, electromagnetism, thermodynamics, relativity and quantum physics explained.

• How Much Does a Smartphone Battery’s Mass Increase When Charged?
  A charged smartphone battery is slightly heavier: estimate the change from its stored energy using Δm = ΔE/c², typically well below a nanogram.
• How to Calculate the Induced EMF of a Spinning Metal Rod
  For a conducting rod rotating in a uniform magnetic field, derive the motional emf, determine its polarity, and work through a numerical example.
• Inductor Charging and Discharging: Current and Voltage
  Learn how an inductor charges and discharges in an RL circuit, including the current, voltage, time constant, and stored energy.
• How to Calculate the Equivalent Capacitance of an Infinite Ladder
  For an infinite ladder of identical capacitors, self-similarity gives an equivalent capacitance of (√5−1)C/2 for the standard topology.
• How Do You Apply Initial Conditions to Simple Harmonic Motion?
  Find the constants in a simple harmonic motion equation from its initial position and velocity, with phase conventions and a worked example.
• What Does the Integral of Momentum with Respect to Time Mean?
  The time integral of momentum is not energy or impulse. For constant mass, it equals mass multiplied by displacement over the interval.
• Bead on a Rotating Hoop: Equilibrium and Stability
  Derive the motion, equilibrium positions, and stability of a frictionless bead on a hoop rotating about its vertical diameter.
• Functional Analysis for Quantum Mechanics: What You Need to Know
  Functional analysis supplies quantum mechanics with Hilbert spaces, self-adjoint operators, spectra, and the rigorous treatment of measurement and evolution.
• How to Interpret a Graph of Electric Potential
  Learn how to read electric-potential graphs, calculate electric field from their slope, and interpret straight, curved, flat, and discontinuous regions.
• How Does Gravity Change With Altitude?
  Gravity decreases with altitude according to the inverse-square law: calculate it with g(h) = g₀[R/(R + h)]² above Earth.
• Does Hydrostatic Pressure Act in Lateral Directions?
  In a fluid at rest, pressure is the same in every direction at a point; lateral force comes from the wall’s normal and varies with depth.
• How to Calculate the Voltage Across a Resistor
  Calculate voltage across a resistor with Ohm’s law, and learn when a resistor gets the full battery voltage or only part of it.
• If Atoms Are Mostly Empty Space, Why Doesn’t Light Pass Through Everything?
  Atoms are not hard little obstacles, but their electrons interact with light. A material’s structure and energy levels determine whether it transmits, absorbs.
• Would a Car’s Headlights Become Gamma Rays Near Light Speed?
  A headlight remains visible in the car’s rest frame, but a forward observer sees relativistic blueshift; calculate the speed needed for gamma rays.
• How Can You Temporarily Demagnetize a Magnet?
  Temporarily reduce a magnet’s field with an opposing electromagnet. Learn why this is reversible only within limits and what methods to avoid.
• How to Measure the EMF of a Cell
  Measure a cell’s EMF with a high-resistance voltmeter for a quick estimate or a potentiometer for a null-balance measurement.
• Hamilton’s Principal and Characteristic Functions: What’s the Difference?
  Hamilton’s principal function S depends on position and time; for a time-independent system, its characteristic function W satisfies a fixed-energy equation.
• How Does Hamilton’s Principle Work with Semiholonomic Constraints?
  For nonintegrable velocity constraints, distinguish Hamilton’s principle on constrained paths from the d’Alembert variation used for ideal constraints.
• Has the Speed of Light Changed Over Time?
  The vacuum speed of light is exactly 299,792,458 m/s by SI definition; whether it varied cosmologically is a separate, dimensionless, unconfirmed question.
• What Can You Use Instead of the Time-Averaged Poynting Vector?
  For periodic fields, use the cycle-averaged Poynting vector; for pulses, integrate instantaneous flux over time. The right choice depends on what you need.
• How Can You Imagine Curled-Up Dimensions?
  Picture a curled-up dimension as a tiny circular direction at every point in space, like the cross-section of a hose seen from far away.
• How Do You Combine Three Resistors in Parallel?
  For three resistors connected across the same two nodes, add their reciprocals, then invert the sum to get the equivalent resistance.
• How Do Mirrors Work?
  Mirrors reflect light because charges in their surfaces respond to incoming electromagnetic waves and emit fields that form a reflected wave.
• How Does a Resistor “Know” to Increase the Voltage Across It?
  A resistor does not sense or choose its voltage: the circuit’s source, resistance, current, and electric fields determine the potential difference.
• How Does Adiabatic vs. Isothermal Compression Affect Gas Work?
  For the same ideal gas and compression ratio, reversible adiabatic compression requires more work than isothermal compression because the gas heats up.
• How Does Electrical Grounding Work?
  Electrical ground can set a circuit's voltage reference or protect against faults. Learn how current returns to its source and when earth matters.
• How Does Pressure Affect the Expansion of the Universe?
  In cosmology, pressure contributes to gravity: positive pressure slows expansion, while sufficiently negative pressure can accelerate it.
• How Does Resistance Relate Voltage to Current?
  Resistance does not create current; it relates current to voltage. Use Ohm’s law for an ohmic resistor and include the rest of the circuit.
• Why Is Current the Same in a Series Circuit?
  In a steady series circuit, charge conservation makes the current equal through every component; voltage drops differ according to each component’s resistance.
• How Does a Cell’s Internal Resistance Vary With Temperature?
  A cell’s internal resistance usually falls as temperature rises because its electrolyte conducts better, but the trend depends on chemistry and conditions.
• Why Does Every Moving Observer Measure the Same Speed of Light?
  A moving observer still measures light in vacuum at c because Lorentz transformations change space and time together, preserving light’s speed.
• How Can a Potential Divider Set a Component’s Voltage to Zero?
  A potential divider can set a load’s voltage to zero by placing its wiper at ground; a series variable resistor usually only approaches zero current.
• How Is the Second Covariant Derivative of a Scalar Computed?
  Compute a scalar's second covariant derivative by differentiating its gradient as a covector and subtracting the Christoffel-symbol term.
• How Isolated Must a System Be to Avoid Wave-Function Collapse?
  There is no universal isolation threshold for avoiding collapse: interactions cause decoherence, while whether that is actual collapse depends on the quantum.
• How to Find the Resonant Amplitude of a Forced Oscillator
  Find a forced oscillator’s resonant amplitude from its mass, damping, stiffness, and drive frequency, and distinguish damped from undamped resonance.
• Given Constant Temperature, Why Can Pressure Affect Internal Energy?
  At constant temperature, ideal-gas internal energy stays fixed, but pressure can change a real fluid’s volume and intermolecular potential energy.
• What Is the Gravitational Constant in Higher Dimensions?
  In D spacetime dimensions, Newton’s constant has dimension-dependent units; compactifying extra dimensions relates it to the observed four-dimensional constant.
• What Is the Magnetic Field Outside a Solenoid?
  For a finite solenoid, the outside field depends on position: use the axial formula on the axis and Biot–Savart integration off axis.
• Can a Monochromatic Pulse Exist?
  A strictly monochromatic wave cannot be a finite pulse: localization in time requires a range of frequencies, though that range can be very narrow.
• Does Matter Become Energy at the Speed of Light?
  Matter does not turn into energy at light speed: massive objects cannot reach that speed, and their rest mass remains part of their total energy.
• Does an Object’s Vibration Frequency Stay Constant in a Vacuum?
  An object's natural vibration frequency usually changes little in a vacuum, but removing air can alter damping and effective mass.
• Does Time Dilation Cause Gravity? What General Relativity Says
  Gravitational time dilation is linked to gravity and can help describe it, but it is not the whole cause: gravity is spacetime geometry.
• Earth Curvature and Refraction, Explained Simply
  Estimate Earth’s curvature drop with distance, see how atmospheric refraction changes it, and use a simple horizon formula with worked examples.
• How Does the Water-Flow Analogy Explain Electric Circuits?
  The water-flow analogy links voltage to pressure, current to flow rate, and resistance to pipe restriction, while clarifying where the comparison fails.
• Why Does Gravity Hold the Planets in Orbit if Electric Forces Are Stronger?
  Electric forces between particles can dwarf gravity, but stars and planets are nearly neutral; their mass adds up, so gravity governs their orbits.
• Does Electricity Take the Path of Least Resistance?
  Electricity does not choose only the least-resistance route. Learn how current divides between branches and how circuit laws determine each path.
• How Are Electromagnetic Radiation and Blackbody Radiation Related?
  Blackbody radiation is the electromagnetic spectrum emitted by an ideal absorber; temperature sets its shape, while Planck’s law and Wien’s law quantify it.
• How to Calculate the EMF Induced in a Moving Rod
  Find the motional EMF in a moving rod using the magnetic force on charges, determine its polarity, and apply the result to a circuit.
• How Are Entanglement and the Pauli Exclusion Principle Related?
  Entanglement and Pauli exclusion are distinct: fermion antisymmetry restricts allowed states, while entanglement describes correlations within a state.
• How Does Diffraction-Grating Spectrometry Work?
  Learn how a diffraction grating separates light by wavelength and how to calculate a spectral line’s wavelength from its order and angle.
• What Is the Divergence of the Gravitational Field?
  In Newtonian gravity, the divergence of the gravitational field is -4πGρ; it vanishes in empty space and is singular at point masses.
• Do Bats Use Special Relativity for Echolocation?
  Bats echolocate with sound in air, not relativistic light signals; classical acoustics and Doppler effects matter, while special-relativity corrections are negligible.
• Do Echolocating Bats Experience the Terrell Effect?
  Bats do not experience the relativistic Terrell effect, though sound-travel delays and Doppler shifts can influence their echo perception.
• Do Individual Photons Gain Entropy as They Travel?
  A photon has no fixed entropy just from traveling: entropy belongs to its quantum state and can change locally through mixing or interactions.
• Do Quantum Energy Generators Produce More Energy Than They Use?
  Quantum energy generators have not been shown to create net energy. Learn how conservation laws and careful input-output tests assess such claims.
• Do Resistor-Based Fan Regulators Save No Power at All?
  A resistor fan regulator wastes power as heat, but the fan may use less at lower speed, so total consumption can still fall.
• Do the Laws of Physics Apply to Non-Observers?
  Relativity’s laws do not require a conscious observer: they apply to physical events, while reference frames describe how those events are measured.
• Is Gravity 9.8 m/s² at Every Height?
  Gravity is about 9.8 m/s² near Earth's surface, but decreases with altitude according to the inverse-square law; learn when the approximation works.
• Does ABS Shorten a Car’s Stopping Distance?
  ABS often shortens braking distance for ordinary drivers, especially on slippery roads, but it does not always beat expert threshold braking or stop sooner on loose.
• Does Current Run Forever in Water?
  Current can continue through water while a complete circuit and electrode reactions are sustained, but real cells eventually change or run out of reactants.
• Does Dark Energy Have Gravity?
  Dark energy gravitates through its energy density and negative pressure: in general relativity it drives accelerated cosmic expansion, though its local effects are tiny.
• Does Earth Have a Code Name?
  Earth has no single universal code name. Its name and technical identifiers depend on whether you mean astronomy, spacecraft software, or fiction.
• Does Gravity Slow the Expansion of the Universe?
  Gravity from matter and radiation decelerates cosmic expansion, but dark energy now causes the universe’s expansion to accelerate.
• Does Light Itself Experience Time Dilation?
  Light has zero proper time along its null path, but it has no rest frame, so it is not meaningful to ask what a photon experiences.
• Can You Store AC in a Battery?
  Batteries store energy chemically, not AC or DC current. Learn why they charge with DC and how converters let AC power charge them.
• Does a Capacitor Pass DC Current?
  An ideal capacitor carries current while its voltage changes, such as during DC charging, but carries no current after steady state is reached.
• What Is Equivalent Resistance?
  Equivalent resistance is the resistance a single resistor would need to match a network’s voltage–current behavior at specified terminals.
• What Is the Pole of a Spherical Mirror?
  The pole of a spherical mirror is the point where its principal axis meets the reflecting surface, usually at the center of the mirror’s aperture.
• Density vs. Mass Density: What Is the Difference?
  Density usually means mass per unit volume in basic physics, but the term also applies to other quantities; context and units clarify which one.
• How to Derive the Expression for Berry Curvature
  Derive Berry curvature from a normalized eigenstate, then express it as a sum over other bands with energy denominators and matrix elements.
• What Is the Derivative of Kinetic Energy?
  The derivative of kinetic energy depends on the variable: with respect to time it is power, and with respect to velocity it is momentum.
• How to Derive a Thermodynamic Relation from Helmholtz Free Energy
  Starting from Helmholtz free energy, derive pressure and entropy identities and show why (∂U/∂V)T = T(∂P/∂T)V − P.
• Can You Determine Whether a Theory Is Unitary from Its Lagrangian?
  A Lagrangian can reveal warning signs such as ghosts, but deciding unitarity also requires the quantization, physical states, and quantum consistency checks.
• What Is the Difference Between I²R, V²/R, and VI for Power?
  VI is the general circuit-power formula; I²R and V²/R come from Ohm’s law for a resistor, using its voltage and current.
• What Happens When Different Batteries Are Connected in Parallel?
  Different-voltage batteries in parallel can drive current into each other. Learn how to estimate that current and why mismatched batteries can be unsafe.
• How to Derive and Solve the Differential Equation for an RC Capacitor
  Derive the RC capacitor differential equation from Kirchhoff’s laws, solve it for charging or discharging, and work through a numerical example.
• How to Convert Joules to Watts
  Joules measure energy and watts measure power. Divide energy in joules by elapsed time in seconds to find average power in watts.
• Can You Convert a Bullet’s Kinetic Energy into Electricity?
  A bullet’s kinetic energy can in principle become electrical energy, but safely capturing it is difficult and real systems lose much of the energy.
• Could the Speed of Light Vary While Time Is Absolute?
  Special relativity makes vacuum light speed invariant and time frame-dependent. An absolute-time model is possible.
• Could Submarine Sonar Kill a Diver?
  A submarine’s active sonar could harm a diver at close range, but ordinary sonar is not known as a reliable way to kill one.
• Could Nuclear Bombs Be Used to Generate Electricity?
  A nuclear bomb could release heat that drives a turbine, but turning an explosion into reliable electricity is far less practical than using a controlled reactor.
• How Are Coulomb’s Law and Electric Field Intensity Related?
  Coulomb’s law gives the force between charges; electric field intensity is the force per unit positive test charge at a point.
• What Are Calabi–Yau Moduli Fields?
  Calabi–Yau moduli fields are four-dimensional scalars describing internal geometry; their number and multiplets depend on Hodge numbers and string theory.
• Can Light and Darkness Coexist?
  Light and darkness can appear together as illumination varies across a scene; darkness is low or absent visible light, not a separate substance.
• How Is MPS Bond Dimension Related to Entanglement?
  An MPS bond of dimension D limits the Schmidt rank across its cut to D and the entanglement entropy to log D, but need not saturate either bound.
• What Is Constructive Interference of Electromagnetic Waves?
  Constructive interference occurs when coherent electromagnetic fields arrive in phase; learn how phase, path difference, polarization, and amplitude determine intensity.
• Can the Mass of a Body Change?
  A body's mass can change when matter or energy enters or leaves it. Learn how classical mass, rest mass, and mass–energy equivalence differ.
• Can the Inner Surface Charge Density Be Negative?
  Yes. A positive charge inside a spherical conducting cavity induces negative charge on the inner wall; mixed-sign interior charges can make the density change sign.
• Can You Change the Capacitance of a Capacitor?
  You can change a capacitor’s capacitance by changing its plate geometry or dielectric; charge and voltage then respond according to its circuit.
• Can You See a Rainbow on the Moon or Other Worlds?
  A lunar surface has no ordinary rainbow because it lacks rain and a substantial atmosphere. Learn what a rainbow needs and where one might occur elsewhere.
• Can Conservative Forces Change a System’s Mechanical Energy?
  A conservative force can change kinetic energy, but its work is offset by an equal change in potential energy when no nonconservative work acts.
• What Is Chemical Potential in Thermodynamics?
  Chemical potential is the free-energy cost of adding matter. Learn its definitions, units, equilibrium role, and an ideal-gas example.
• How to Draw a Free-Body Diagram for Circular Motion
  For circular motion, draw only real forces; their inward radial components must add to mv²/r. See worked diagrams for turns and strings.
• How to Calculate a Voltmeter Reading
  Calculate a voltmeter reading by identifying its connection points, finding the voltage difference between them.
• Can a Liquid Boil in a Closed Container?
  A liquid can boil in a closed container, but heating usually raises the pressure too, changing its boiling temperature and creating a rupture hazard.
• Can the Focal Length of a Convex Lens Be Negative?
  A convex lens usually has positive focal length in air, but it can have negative focal length if its surroundings make it diverge light.
• Can Internal Forces Change a System’s Total Momentum?
  Internal forces can redistribute momentum among a system’s parts, but they cannot change its total momentum when the system is isolated.
• Can Magnitude Be Negative?
  Magnitude is never negative: it measures size. Signed quantities such as potential energy or a vector component can be negative, depending on direction or reference.
• Can Planck’s Constant Be Derived From Maxwell’s Equations?
  Maxwell’s equations predict classical electromagnetic waves, not Planck’s constant; quantum theory supplies the action scale needed to describe photon energies.
• Can a Digital Computer Simulate Quantum Entanglement to Any Precision?
  A classical digital computer can approximate entangled quantum systems to arbitrary precision in principle, but general simulations can require exponential resources.
• Can Electrical Resistance Depend on Direction?
  Ordinary linear resistors have the same resistance when current reverses, but anisotropic materials and nonlinear devices can behave differently.
• What Is Planck’s Constant, Simply Explained?
  Planck’s constant sets the scale of quantum effects and relates a photon’s energy to its frequency through the equation E = hf.
• Can the Accuracy of Planck’s Constant Be Increased?
  Planck’s constant has an exact numerical value in the SI since 2019; experiments can improve its practical realization, not its defined value.
• What Is a Voltage Drop in a DC Circuit?
  Learn what voltage drop means in a DC circuit, how to calculate it with Ohm’s law, and how drops behave in series and parallel circuits.
• What Are Bound Charges in a Dielectric Material?
  Bound charges arise from dielectric polarization: their volume density is minus the divergence of polarization, and their surface density is its outward normal component.
• How to Calculate Acceleration and Time from Speed and Distance
  Find acceleration and travel time from initial speed, final speed, and distance using constant-acceleration equations, with a worked example and sign checks.
• How to Calculate Field of View From Focal Length
  Focal length alone does not determine field of view: use the sensor dimension and calculate 2 arctan(sensor size divided by 2 focal length).
• How Much Ice Is Needed to Cool Water by a Given Temperature?
  Calculate the ice mass needed to cool water by a chosen amount using a heat balance that includes melting and warming the melted ice.
• How Is Angular Displacement Related to the Displacement Vector?
  Angular displacement measures rotation, while the displacement vector joins initial and final positions; on a circle, relate them using arc and chord formulas.
• Are Coherent States of Light Classical or Quantum?
  Coherent states are quantum states with classical-like field behavior, Poisson photon statistics, and unavoidable quantum fluctuations in measurements.
• Are Electromagnetic Waves the Only Way to Transmit Information?
  No. Sound, mechanical motion, particles, and other physical signals can carry information; relativity limits causal signaling to at most light speed in vacuum.
• What Are the Arguments For and Against Many-Worlds?
  Many-worlds keeps the quantum state evolving without collapse, but its costs include explaining probability, branching, and its vast ontology.
• What Happens Atomically When Batteries Are Connected in Series?
  In a series connection, each battery’s chemical reactions contribute energy per unit charge, so aligned emfs add to give a larger total voltage.
• At What Speed Is the Universe Expanding?
  The universe has no single expansion speed: its current rate is about 70 km/s per megaparsec, so more distant galaxies recede faster.
• What Are the Applications of the Wave Equation?
  The wave equation models how disturbances travel. Explore its uses in strings, sound, light, water, and seismic waves, with a worked string example.
• What Is the Area Under a Displacement–Time Graph?
  For a displacement–time graph, integrate displacement over time: the result has units of distance × time, not displacement. Learn how to calculate it.
• How Does Boltzmann’s Transformation Reduce the Heat Equation?
  See how Boltzmann’s similarity variable reduces the heat equation to an ODE, with the chain-rule derivatives and an error-function example.
• How to Calculate Average Acceleration Without Time
  Find acceleration without time from initial and final velocities and displacement when acceleration is constant; otherwise, more information is needed.
• How to Calculate the Acceleration of a Moving Elevator
  Calculate elevator acceleration from its change in velocity or from the net force, and determine whether it accelerates upward or downward.
• How to Calculate Net Force from a Free-Body Diagram
  Resolve every force on the free-body diagram into components, add signed components, then combine them to find the net force's magnitude and direction.
• How Is the Fermi–Dirac Distribution Derived?
  The Fermi–Dirac distribution follows by counting allowed fermion occupations and maximizing multiplicity: f(ε)=1/[exp((ε−μ)/kT)+1].
• How Is the Lagrangian Derived in Classical Mechanics?
  Derive the Euler–Lagrange equations from the stationary-action principle, see why L = T − V works, and follow a worked spring example.
• Did Space and Time Exist Before the Big Bang?
  The Big Bang model does not establish what came before: classical relativity reaches a boundary, while quantum-gravity proposals remain unconfirmed.
• What Is the Electric Field Around a Current-Carrying Conductor?
  A steady current needs a longitudinal electric field inside a resistive wire; the field outside depends on surface charges, voltage, and the return-path geometry.
• What Is the Fourier Series of a Square Wave?
  For a symmetric square wave of amplitude ±A and period T, the Fourier series contains only odd sine harmonics, with coefficients decreasing as 1/n.
• How Does a Charged Particle Move in a Magnetic Field?
  In a uniform magnetic field, a charged particle follows a helix when its velocity has parallel and perpendicular components; calculate its radius and pitch.
• How Do Astronomers Identify Elements in a Blended Emission Spectrum?
  Astronomers identify elements in a blended emission spectrum by matching calibrated line wavelengths to atomic data and checking shifts, blends, and line strengths.
• How to Calculate Engine Torque from Wheel Torque
  Recover engine torque from wheel torque by dividing by transmission and final-drive ratios and drivetrain efficiency, while checking wheel or axle readings.
• How to Calculate a Viscous Damping Coefficient
  Calculate a viscous damping coefficient from force and velocity, mass and stiffness, or measured oscillation decay, with units and a worked example.
• How to Reveal Electric Field Lines with Grass Seeds in Mineral Oil
  Use polarized grass seeds in insulating mineral oil to trace electric-field direction; learn the setup, what the pattern means, and how to work safely.
• Tensor Product vs. Cartesian Product in Quantum Systems
  For two quantum subsystems, the joint Hilbert space is a tensor product, not a Cartesian product of state vectors; see dimensions, product states, and entanglement.
• Tensor Product vs. Direct Product for Three Spin-1/2 Particles
  For three spin-1/2 particles, the composite Hilbert space is an eight-dimensional tensor product; total-spin sectors give a direct-sum decomposition.
• What Is the Color of a Mirror?
  A mirror is usually described as silver or neutral gray, but its apparent color comes from reflected light and can show a slight green tint.
• What Is the Relationship Between Light Intensity and Wavelength?
  Light intensity is not set by wavelength alone: at fixed wave amplitude it is independent, while at fixed photon flux it scales as 1/λ.
• What Is the Difference Between a Particle and a Wave?
  A particle is localized; a wave is a spreading disturbance. Learn the classical difference and why quantum objects show both particle-like and wave-like behavior.
• How Is the Strong Equivalence Principle Stronger Than the Weak One?
  The strong equivalence principle extends universal free fall to self-gravitating bodies and gravitational experiments; here is how it differs from the weak principle.
• Why Does Water Feel Cooler Than Air at the Same Temperature?
  Water usually feels colder than air at the same temperature because it removes heat from skin faster; conductivity, heat capacity, and evaporation explain why.
• Why Is Entropy an Extensive Quantity?
  Entropy is extensive because independent subsystems have multiplying numbers of microstates, and the logarithm turns that product into a sum.
• Would a Foucault Pendulum Work on a Rotating Flat Earth?
  A Foucault pendulum would work on an ideal rotating flat surface: its swing plane would precess opposite the rotation, at about one turn per day.
• How to Calculate the Phase Angle (Circuits, Phasors and Waveforms)
  Phase angle formulas for series RLC circuits (φ = arctan((X_L − X_C)/R)), complex phasors, and two sine waves from a time shift, with worked examples.