Study sheet: Electromagnetic Waves and Optics

📋 Course Outline

1. Scalars and Vector Quantities
2. Vector Representation and Operations
3. Position Displacement and Velocity
4. Average and Instantaneous Velocity
5. Acceleration and Motion Equations
6. Motion Graphs and Relative Velocity
7. Elasticity and Plasticity
8. Density and Elastic Moduli
9. Static Equilibrium and Torque
10. Static Equilibrium Conditions
11. Electric Charge and Charging
12. Fields, Circuits, and Current
13. Ohm’s Law and Resistance
14. Series and Parallel Circuits
15. Meters, Safety, and Projects
16. Magnets and Magnetic Fields
17. Earth and Current Magnetic Fields
18. Magnetic Forces and Applications

📖 1. Scalars and Vector Quantities

🔑 Key Concepts & Definitions

• Scalar quantity : a physical quantity completely specified by a single number and an appropriate unit of measurement
• Vector quantity : a physical quantity specified by both magnitude and direction

📝 Essential Points

• Examples of scalar quantities include:

  • time
  • distance
  • speed
  • length
  • volume
  • temperature
  • energy
  • power

• Examples of vector quantities include:

  • displacement
  • velocity
  • acceleration
  • momentum
  • impulse
  • weight
  • force
  • electric field strength

💡 Memory Hook

Scalars have magnitude only, whereas vectors have magnitude and direction.

📖 2. Vector Representation and Operations

🔑 Key Concepts & Definitions

• Vector magnitude : The magnitude of a vector is represented by the length of its arrow when the vector is drawn to scale, while the arrowhead represents its direction.
• Resultant vector : the single vector whose effect is the same as the combined effect of the individual vectors
• Vector resolution : the process of breaking a vector into component vectors whose sum equals the original vector

★ Must-know

• 🔄 To draw a vector graphically: choose and record a scale, determine the arrow length from the scale, draw the arrow in the required direction with an arrowhead, label its magnitude

📐 Formula — The resultant of vectors A and B is $\mathbf{R}=\mathbf{A}+\mathbf{B}$.

📐 Formula — Subtracting vector B from vector A means adding the negative of B: $\mathbf{A}-\mathbf{B}=\mathbf{A}+(-\mathbf{B})$.

📐 Formula — For a vector A making an angle θ with the positive x-axis, its rectangular components are $A_x=A\cos\theta$ and $A_y=A\sin\theta$.

Further detail

• The main vector types are: a zero vector with zero magnitude and no direction, a unit vector with magnitude one, equal vectors with the same magnitude and direction, the negative of a vector with the same magnitude but opposite direction

• 🔄 The triangle method: place the head of each successive vector at the tail of the next vector, join the tail of the first vector to the head of the last vector, use the joining vector as the resultant

📐 Formula — For perpendicular vectors A and B, the resultant magnitude is $|\mathbf{R}|=\sqrt{A^2+B^2}$ and its direction satisfies $\theta=\tan^{-1}\left(\frac{B}{A}\right)$.

💡 Memory Hook

Represent → add → subtract → resolve.

📖 3. Position Displacement and Velocity

🔑 Key Concepts & Definitions

• Position : the location of an object relative to an origin or reference frame and may be positive or negative
• Average velocity : the displacement of a body divided by the time interval during which that displacement occurs

★ Must-know

📌 Distance is the total length of the path traveled and is a scalar, whereas displacement is the difference between final and initial positions and is a vector.

📐 Formula — If an object moves from initial position s₀ to final position s, its displacement is $\Delta s=s-s_0$.

📐 Formula — The average velocity of an object is $v_{av}=\frac{\Delta s}{\Delta t}=\frac{s-s_0}{t-t_0}$.

Further detail

• For motion 60 km from O to A and then 25 km back toward B, the distance traveled is 85 km while the displacement magnitude is 35 km.

• For a person walking 70 m east and then 30 m west, the displacement is 40 m east and the total distance traveled is 100 m.

💡 Memory Hook

Distance follows the path, whereas displacement joins initial and final positions.

📖 4. Average and Instantaneous Velocity

🔑 Key Concepts & Definitions

• Instantaneous velocity : the velocity of a body at a specific instant or over an infinitesimally small time interval

★ Must-know

📐 Formula — Average velocity is calculated as $v_{av}=\frac{\Delta s}{\Delta t}=\frac{s-s_0}{t-t_0}$, while average speed is calculated as $v_{av}=\frac{s_{tot}}{t_{tot}}$.

Further detail

• The magnitude of a car’s instantaneous velocity is the reading of its speedometer.

• For motion in the same direction along a straight line, average speed equals the magnitude of average velocity.

💡 Memory Hook

Average velocity describes an interval, whereas instantaneous velocity describes one instant.

📖 5. Acceleration and Motion Equations

🔑 Key Concepts & Definitions

• Acceleration : the rate of change of velocity with time and occurs when an object speeds up, slows down, or changes direction
• Uniformly accelerated motion : straight-line motion in which velocity increases or decreases by equal amounts in equal time intervals

★ Must-know

📐 Formula — Average acceleration is calculated as $a=\frac{\Delta v}{\Delta t}=\frac{v-v_0}{t-t_0}$ and has the SI unit $\mathrm{m/s^2}$.

📌 When an object speeds up, acceleration points in the direction of motion, whereas when it slows down, acceleration points opposite to the direction of motion.

📐 Formula — For constant acceleration, the equations of motion are $v=v_0+at$, $s=v_0t+\frac{1}{2}at^2$, and $v^2=v_0^2+2as$.

Further detail

• Near Earth’s surface, the free-fall acceleration is approximately $9.80\,\mathrm{m/s^2}$ downward when air resistance is neglected.

💡 Memory Hook

A change in velocity causes acceleration, which determines uniformly accelerated motion.

📖 6. Motion Graphs and Relative Velocity

🔑 Key Concepts & Definitions

• Relative velocity : the velocity of one object with respect to another object or observer

★ Must-know

• The graph shapes are:
  • parabolic position–time graph
  • straight-line velocity–time graph
  • time-axis-parallel acceleration–time graph

📌 The slope of the tangent to a position–time graph gives instantaneous velocity.

📌 The slope of a velocity–time graph gives acceleration, and the area under a velocity–time graph gives displacement.

📐 Formula — For objects A and B, relative velocities are $v_{AB}=v_A-v_B$ and $v_{BA}=v_B-v_A$; objects moving in opposite directions have a relative-speed magnitude equal to the sum of their speeds, while objects moving in the same direction have the difference of their speeds.

Further detail

📌 The area under an acceleration–time graph gives the change in velocity.

💡 Memory Hook

Position–time slope gives velocity; velocity–time slope gives acceleration; graph areas give displacement or velocity change.

📖 7. Elasticity and Plasticity

🔑 Key Concepts & Definitions

• Elasticity : the property of a body or material that allows it to regain its original shape and size after the deforming force is removed
• Plasticity : the ability of a body or material to undergo permanent and irreversible deformation after the deforming force is removed
• Elastic Limit : the maximum deforming force up to which a body retains its elastic behavior and returns to its original state

📝 Essential Points

• A deforming force is an external force required to change the shape or size of a body.

• Applications of elasticity include: keeping machine parts below the elastic limit, selecting crane-rope thickness using the elastic limit and factor of safety, explaining why long-used bridges can become unsafe through repeated strains, estimating the maximum height of a mountain from Earth’s elastic behavior

💡 Memory Hook

Elasticity returns to the original shape; plasticity leaves permanent deformation.

📖 8. Density and Elastic Moduli

🔑 Key Concepts & Definitions

• Density : the mass of a substance per unit volume and is calculated by $\rho=\frac{m}{V}$
• Specific Gravity : the ratio of the density of a substance to the density of a standard substance, commonly water at 4 °C, and is dimensionless
• Stress : the deforming force per unit area and is calculated by $\sigma=\frac{F}{A}$, with SI unit N/m²
• Strain : the fractional deformation produced in an object under stress and is dimensionless
• Young Modulus : the ratio of tensile or compressive stress to longitudinal strain, calculated by $Y=\frac{\text{tensile stress}}{\text{tensile strain}}$

★ Must-know

📐 Formula — Specific gravity is calculated by $SG=\frac{\rho_{\mathrm{substance}}}{\rho_{\mathrm{water}}}$.

Further detail

• The SI unit of density is kg/m³, and the relation between units is $1\,\mathrm{kg/m^3}=10^{-3}\,\mathrm{g/cm^3}$.

📐 Formula — The three strain relations are tensile or compressive strain $\frac{\Delta L}{L_0}$, volumetric strain $\frac{\Delta V}{V_0}$, and shearing strain $\frac{\Delta x}{L_0}$.

💡 Memory Hook

Force per area causes stress, deformation produces strain, and their ratio gives an elastic modulus.

📖 9. Static Equilibrium and Torque

🔑 Key Concepts & Definitions

• Static Equilibrium : occurs when a body remains at rest without tilting or rotating, with both its net force and net torque equal to zero
• Torque : Torque is the twisting effect of a force about a pivot and has magnitude $\tau=Fr\sin\theta$, or $\tau=Fr$ when the force is perpendicular to the lever arm.

★ Must-know

📌 The first condition of equilibrium is that the vector sum of all forces is zero, expressed as $\sum \vec{F}=0$; in two dimensions this requires $\sum F_x=0$ and $\sum F_y=0$.

📌 The second condition of equilibrium is that the resultant external torque about an axis is zero, expressed as $\sum\tau=0$, so clockwise and counterclockwise torques balance.

Further detail

• A standard equilibrium solution proceeds by drawing a free-body diagram, choosing coordinates and resolving forces, writing $\sum F_x=0$ and $\sum F_y=0$, writing $\sum\tau=0$ about a convenient axis, and solving for the unknowns.

• For a seesaw, a 30 kg child sitting 2.5 m from the pivot is balanced by a 25 kg child sitting 3.0 m from the pivot because their opposite torques are equal.

💡 Memory Hook

F–τ: zero net force prevents translation, zero net torque prevents rotation.

📖 10. Static Equilibrium Conditions

🔑 Key Concepts & Definitions

• Static equilibrium : occurs when an object or system remains at rest without tilting or rotating.
• Torque : the product of the distance from a support or pivot and the component of force perpendicular to the object.

★ Must-know

📌 The first condition of equilibrium is that the vector sum of all external forces acting on a body is zero, $\sum \vec{F}=0$.

📌 The second condition of equilibrium is that the sum of all torques calculated about any arbitrary axis is zero, $\sum \tau=0$.

Further detail

• For the seesaw example, torque balance gives $m_Ag(2.5\,\mathrm{m})-m_Bg(x)=0$, so a 30 kg child balances a 25 kg child when the second child sits 3.0 m from the pivot.

💡 Memory Hook

Force balance prevents translation, whereas torque balance prevents rotation.

📖 11. Electric Charge and Charging

🔑 Key Concepts & Definitions

• Electric charge : a property that occurs as positive charge or negative charge; protons are positive, electrons are negative, and neutral objects contain equal amounts of both.
• Electroscope : a sensitive device used to detect the presence and type of electric charge, identify conductors and insulators, and measure the quantity of charge.

★ Must-know

📌 The law of conservation of charge states that the total electric charge in an isolated system never changes because charge is transferred rather than created or destroyed.

📐 Formula — Electric charge is quantized according to $q=ne$, where $n$ is a positive or negative integer and $e$ is the elementary charge magnitude.

• The three methods of charging a body are:
  • charging by rubbing
  • charging by conduction
  • charging by induction

📌 Charging by conduction leaves the charged and initially uncharged bodies with the same sign of charge, whereas charging by induction leaves the uncharged body with the opposite sign of charge.

Further detail

• Electrical discharge is the rapid transfer of electric charges, and discharging a charged body through a conductor makes it neutral.

💡 Memory Hook

Rubbing → conduction → induction → detection → discharge

📖 12. Fields, Circuits, and Current

🔑 Key Concepts & Definitions

• Electric field : a region in which an electric charge experiences an electric force.
• Electric circuit : a path through which electric charges can flow and consists in its simplest form of a source, conducting wires, and a load.
• Potential difference : Alessandro Volta, 1745-1827 — Potential difference between two points is the work done to move a unit charge from one point to the other, $V=\frac{W}{Q}$, and is measured in volts.

★ Must-know

📐 Formula — Coulomb's law gives the magnitude of the electrostatic force between two point charges as $F=k\frac{|q_1||q_2|}{r^2}$, where $k\approx9\times10^9\,\mathrm{N\,m^2/C^2}$. — Charles Coulomb, 1785

📌 The electrostatic force acts along the line joining two charges, is attractive for unlike signs, and is repulsive for like signs.

📐 Formula — Electric field strength is the force per unit positive test charge, $E=\frac{F}{q}$, and its SI unit is newton per coulomb.

📐 Formula — Electric current is the rate of flow of charge, given by $I=\frac{\Delta Q}{\Delta t}$, with SI unit ampere.

📐 Formula — Ohm's law states that the potential difference across a conductor equals the product of current and resistance, $V=IR$. — Georg Simon Ohm, 19th century

Further detail

📐 Formula — The electric field produced by a point charge at distance r has magnitude $E=k\frac{|q|}{r^2}$ and points outward from a positive charge or inward toward a negative charge.

📌 A closed circuit has a complete conducting path and allows current to flow, whereas an open circuit has a broken path, zero current, and no glowing bulb.

💡 Memory Hook

Potential difference → current; resistance opposes current.

📖 13. Ohm’s Law and Resistance

🔑 Key Concepts & Definitions

• Resistance : the property of a conductor or component that opposes the flow of electric charge and controls the magnitude of the current
• Resistivity : a characteristic property of a material that quantifies its opposition to electric current and has SI unit Ωm

★ Must-know

📐 Formula — Ohm’s law states that the voltage across a conductor equals the product of its current and resistance: $V = IR$. — Georg Simon Ohm, 1787-1854

📌 Ohmic materials have a linear current–voltage relationship and approximately constant resistance over a wide voltage range, whereas non-ohmic materials have a nonlinear relationship.

📐 Formula — The resistance of a uniform conductor is given by $R = \rho\frac{L}{A}$, so it increases with length and decreases with cross-sectional area.

Further detail

• The SI unit of resistance is the ohm, represented by Ω, and $1\,\Omega = 1\,\mathrm{V/A}$.

• A 220 V source connected to a 1200 Ω bulb produces a current of approximately 0.18 A.

💡 Memory Hook

Voltage causes current, while resistance opposes it.

📖 14. Series and Parallel Circuits

🔑 Key Concepts & Definitions

• Series circuit : provides only one path for electric current, so opening one component stops current throughout the circuit
• Parallel circuit : provides multiple conducting paths between common junctions, so current can continue through other branches if one branch opens

★ Must-know

📐 Formula — For resistors in series, the current is the same through every resistor, the voltage is divided among them, and the equivalent resistance is $R_{eq}=R_1+R_2+\cdots+R_n$.

📐 Formula — For resistors in parallel, the voltage is the same across every branch, the total current is the sum of branch currents, and the equivalent resistance satisfies $\frac{1}{R_{eq}}=\frac{1}{R_1}+\frac{1}{R_2}+\cdots+\frac{1}{R_n}$.

Further detail

• 🔄 A series–parallel circuit is analyzed by these steps:

  1. Reduce the combination farthest from the source
  2. Continue until one equivalent resistance remains
  3. Calculate total current with Ohm’s law
  4. Expand the circuit in reverse order to find branch currents and voltages

• Three parallel resistors of 12 Ω, 12 Ω, and 6.0 Ω connected to 12 V have an equivalent resistance of 3.0 Ω, a total current of 4.0 A, and branch currents of 1.0 A, 1.0 A, and 2.0 A.

💡 Memory Hook

Series has one path; parallel has separate paths.

📖 15. Meters, Safety, and Projects

★ Must-know

📌 A voltmeter measures potential difference and is connected in parallel with the component, whereas an ammeter measures current and is connected in series.

📌 A voltmeter has high resistance so that it draws minimal current, whereas an ammeter has very low resistance so that it minimally changes the circuit current.

• Electric-shock damage depends on the current magnitude, the duration of contact, and the body part through which the current passes.

• Currents of 5 mA or less usually cause a shock sensation with little or no damage, currents above about 10 mA can contract hand muscles, and about 100 mA through the body for a few seconds can be fatal.

Further detail

📌 Connecting an ammeter in parallel can make it draw excessive current and become damaged, while connecting a voltmeter in series can greatly increase circuit resistance and stop the current.

• 🔄 A fuse responds to excessive current by:
  1. Heating
  2. Melting
  3. Breaking the circuit

📌 A grounding wire provides a low-resistance path to ground during a fault so that a fuse blows or a circuit breaker trips before the user is injured.

📌 Ground-fault interrupters or residual current devices detect leakage currents of approximately 5 mA or greater and interrupt the current in less than a millisecond.

💡 Memory Hook

A voltmeter goes across; an ammeter goes through.

📖 16. Magnets and Magnetic Fields

🔑 Key Concepts & Definitions

• Magnet : a material or object that produces a magnetic field capable of attracting other materials and attracting or repelling other magnets
• Magnetic field : the region around a magnet or moving electric charge in which magnetic force acts

📝 Essential Points

📌 Permanent magnets retain their magnetic properties after magnetization, temporary magnets lose their magnetic properties when the external magnetic field is removed, and electromagnets produce a magnetic field when current flows through a coil around an iron core.

• Every magnet has a North pole and a South pole; magnetic poles always occur in pairs, so cutting a magnet produces pieces that each have both poles.

📌 Like magnetic poles repel each other, unlike magnetic poles attract each other, and the magnetic force becomes greater as the distance between the magnets decreases.

📌 Magnetic field lines never intersect, form closed loops, point from North to South outside a magnet and from South to North inside it, and are denser where the magnetic field is stronger.

💡 Memory Hook

Like poles repel, unlike poles attract.

📖 17. Earth and Current Magnetic Fields

🔑 Key Concepts & Definitions

• Compass : an instrument containing a magnetized needle that is free to turn and aligns with the direction of a magnetic field

★ Must-know

• The Earth’s magnetic field is generated by the movement of molten iron in the outer core and has a shape similar to that of a bar magnet tilted about 11° from the geographic poles.

📐 Formula — The magnetic field strength at distance d from a long straight wire carrying current I is $B = \frac{\mu_0 I}{2\pi d}$, where $\mu_0 = 4\pi \times 10^{-7}\ \mathrm{T\,m\,A^{-1}}$.

📌 A long straight current-carrying wire produces closed concentric circular magnetic field lines in planes perpendicular to the wire, and reversing the current reverses the field direction.

Further detail

• The SI unit of magnetic field strength is the tesla, and 1 G = 10^{-4} T.

💡 Memory Hook

Moving charges → magnetic fields.

📖 18. Magnetic Forces and Applications

🔑 Key Concepts & Definitions

• Electromagnetic wave : consists of oscillating electric and magnetic fields that radiate outward from a source at the speed of light
• Speed of light : the constant vacuum speed of electromagnetic waves, represented by c and equal to $2.99792458 \times 10^8\ \mathrm{m/s} \approx 3.00 \times 10^8\ \mathrm{m/s}$
• Light ray : a straight line with an arrowhead used to represent the direction and path of light propagation
• Reflection : the change in direction of light rays at a surface that causes them to move away from the surface

📝 Essential Points

📐 Formula — The magnetic force on a charge q moving with speed v in a magnetic field B at angle θ is $F = qvB\sin\theta$.

📐 Formula — The magnetic force on a wire of length L carrying current I in a uniform magnetic field B at angle θ is $F = ILB\sin\theta$.

• The magnetic force on a moving charge or current-carrying wire is perpendicular to the plane formed by the velocity or conductor and the magnetic field, with its direction determined by the right-hand rule.

• Two parallel wires carrying currents in the same direction attract each other, whereas two parallel wires carrying currents in opposite directions repel each other.

• Mechanical waves require matter to transfer energy, whereas electromagnetic waves can travel through empty space or matter.

• Electromagnetic waves are transverse because their electric and magnetic fields oscillate at right angles to the direction of propagation.

• The electromagnetic spectrum consists of:

  • radio waves
  • microwaves
  • infrared
  • visible light
  • ultraviolet
  • X-rays
  • gamma rays

📌 Across the electromagnetic spectrum, increasing frequency corresponds to decreasing wavelength and increasing energy.

• Visible light has frequencies of approximately $4 \times 10^{14}\ \mathrm{Hz}$ to $7 \times 10^{14}\ \mathrm{Hz}$ and vacuum wavelengths of approximately 700 nm to 400 nm.

• Gamma rays have the highest energy and greatest penetrating ability in the electromagnetic spectrum, while radio waves have the lowest energy and least penetrating ability.

📐 Formula — For an electromagnetic wave, the speed of light satisfies $c = \lambda f$, where $\lambda$ is wavelength and $f$ is frequency.

📌 An observer sees an object only when light from the object enters the eyes, either because the object emits light or because it reflects light from a source.

💡 Memory Hook

Charge → wire → parallel wires.

📊 Synthesis Tables

Scalars and vectors

Feature	Scalars	Vectors
Required information	Magnitude only	Magnitude and direction
Examples	Time, distance, speed, temperature	Displacement, velocity, force, acceleration
Graphical representation	Number with unit	Arrow length and arrowhead direction

Motion Graph Interpretations

Graph feature	Physical quantity
Position–time tangent slope	Instantaneous velocity
Velocity–time slope	Acceleration
Velocity–time area	Displacement
Acceleration–time area	Change in velocity