Reflection and refraction Dispersion 6. Oscillation and waves … 6.3 Wave motion Reflection and refraction Dispersion Variation of wave characteristics on the boundary 7. Electrostatic field 14 7.1 Electric charge 7.2 Coulomb’s law 7.3 Electric field in vacuum 7.4 Motion of a charged particle in an electric field 7.5 Electric field in medium Physics I-2019, Lecture 7

6.3 Wave motion from now: I will talk very often on light, however, the laws and rules are valid for a wave motion, generally Physics I-2019, Lecture 7

Reflection and refraction in homogeneous media – the wave propagates with the same velocity in all directions wavefronts – surfaces of constant phase (crest of a water wave) the point source of wave, homogeneous media, then the wavefronts are spherical ray – line perpendicular to the wavefront rays for plane wave - parallels Source ≡ point in very great distance plane wave spherical wavefronts Physics I-2019, Lecture 7

Huygens’s principle: Every point of a wavefront get rise an elemental spherical wavefront -wavelets. Next wavefront is the surface tangent to these wavelets. in hom. media - a spherical wave is propagated as a spherical wave, a plane wave is propagated as a plane wave Physics I-2019, Lecture 7

Laws for propagation of waves 1. Rays in homogeneous media propagates in straight lines. 2. The angle of incidence (made by an incident ray and normal to surface ) equals the angle of reflection (made by a reflected ray and normal to surface ); the incident and reflected rays stay in the plane of incidence. 3. The law of refraction, see further 2 hom. media, wave speeds 𝑣 1 , 𝑣 2 , the boundary surface is a plane, plane wave - angle of incidence 𝛼 between incidence ray and normal - plane of incidence angle of refraction 𝛽 between refracted ray and normal board The law of refraction (Snell’s) - the plane of incidence = plane of reflection = plane of refraction hypotenuse 𝑛 21 … relative index of refraction sin 𝛼 sin 𝛽 = 𝑣 1 𝑣 2 = 𝑛 21

𝑐 … the wave velocity in vacuum The law of refraction 𝑛 21 … relative index of refraction (absolute) index of refraction 𝑛 – relative index from vacuum into substance 𝑐 … the wave velocity in vacuum sin 𝛼 sin 𝛽 = 𝑣 1 𝑣 2 = 𝑛 21 𝑛= 𝑐 𝑣 the law of refraction using index of refraction 𝑛 21 = 𝑛 2 𝑛 1 sin 𝛼 sin 𝛽 = 𝑛 2 𝑛 1 for 𝑛 2 > 𝑛 1 → refraction toward normal for 𝑛 2 < 𝑛 1 → refraction away from normal Physics I-2019, Lecture 7

consider 𝑛 2 < 𝑛 1 → refraction away from normal critical angle ac (am )≡ angle of incidence, where the angle of refraction = 90° a > ac … total internal reflection: air water board sin 𝛼 𝑐 = 𝑛 2 𝑛 1 = 𝑛 21 Physics I-2019, Lecture 7

Optical fiber (fibre) cover cladding core source core cladding Practical application of total internal reflection: The rays which enter the fibre and strike the fibre walls at angles which are greater then critical angler cannot escape from the fibre. The light is therefore channeled along the fibre with only small loses. Major importance in the communication, enable to view sites without minimal disturbance to the patient. core cladding Physics I-2019, Lecture 7

≡ diffraction index varies with wavelength Dispersion ≡ diffraction index varies with wavelength normal dispersion – the index of diffraction decreases as the wavelength increases silica glass prism red light white light blue light application: dispersion according to wavelength Physics I-2019, Lecture 7

Variation of wave phase and wavelength on the boundary i) variation of phase on boundary model situation for light: lower index of ref. → higher index ≡ string with a solid boundary higher index of ref. → lower index ≡ sting with a free loop solid boundary free loop the phase change of p rad upon reflection – “phase reversal” phase unchanged Physics I-2019, Lecture 7

Variation of wave phase and wavelength on the boundary 𝑓= 𝑣 𝜆 Variation of wave phase and wavelength on the boundary ii) variation of wavelength on transmission trough a boundary wavelength in a medium of ref. index n … wavelength in vacuum … frequency – property of source wavelength – property of medium 𝜆 2 𝜆 1 = 𝑣 2 𝑣 1 = 𝑛 1 𝑛 2 𝜆=𝑣𝑇 𝜆= 𝜆 0 𝑛 𝜆 0 =𝑐𝑇 Physics I-2019, Lecture 7

“static” - does not change with time 7.1. Electric charge 7. Electrostatic field “static” - does not change with time 7.1. Electric charge electric charge q, Q connected with a material object (with its carrier) two kinds of charge, > 0, < 0 property of subatomic particles unit: C (coulomb) law: of conservation, of invariance, quantization charged, uncharged, polarized objects a point charge – approximation where the distribution of charge in a body is unimportant = object whose dimensions are negligible and which has a charge │Q│= n e, e = 1,6 . 10-19 C n … integer Physics I-2019, Lecture 7

formulated for point chrages 7.2 Coulomb’s law 14-1 charges of the same sign: 𝑄 1 𝑄 2 >0 – repulsive force, 𝐹 21 ↑↑ 𝑟 21 charges of different sign: 𝑄 1 𝑄 2 <0 – atreactive force, 𝐹 21 ↑↓ 𝑟 21 formulated for point chrages 𝑟 21 = 𝑟 2 - 𝑟 1 = position vector of the point charge Q2 with respect to Q1 𝐹 21 =𝑘 𝑄 1 𝑄 2 𝑟 21 2 𝑟 21 𝑟 21 𝑘= 1 4𝜋 𝜀 0 Coulomb’s law, 𝜀 0 permittivity of free space/vacuum 𝜀 0 =8,85∙ 10 −12 C 2 m −2 N −1 proportionality constant 𝐹 21 = − 𝐹 12 𝐹 21 =𝑘 𝑄 1 𝑄 2 𝑟 12 2 Physics I-2019, Lecture 7

for a group of point charges – the forces are summarized vectorically Example: n point charges Q1, Q2, …, Qn exert on a point charge Q0 𝐹 = 𝑖=1 𝑛 𝐹 0𝑖 = 𝑖=1 𝑛 𝑘 𝑄 𝑖 𝑄 0 𝑟 0𝑖 2 𝑟 0𝑖 𝑟 0𝑖 Physics I-2019, Lecture 7

Q0 – „a test charge“, (small, positive, its filed is negligible) 7.3 Electric/electrostatic field in vacuum 14-2 electrostatic field – the force field which extends outward of each charge, this force field exerts on other charges, e.g on charge Q0: Electric field in the point: Deff. i) The field of a point charge 𝐹 = 𝑄 0 𝐸 the magnitude of 𝐸 is the force per unit positive charge Q0 – „a test charge“, (small, positive, its filed is negligible) unit of 𝐸 : N/C, V/m 𝐸 = 𝐹 𝑄 0 𝐸 =𝑘 𝑄 𝑟 2 𝑟 𝑟 field of the point charge Q at point 𝑟 the magnitude is the same on surface of sphere, the direction is radial the field is not defined in 𝑟 =0 E(r/2) = ?, E(2r) = ? if Q < 0 → 𝐸 of opposite direction the field of a point charge is not homogenous electric field lines - 𝐸 is tangential to the electric filed lines (lines of force)

ii) Electric field of a group of point charges: Physics I-2019, Lecture 7

Electric field of a dipole a dipole – positive and negative point charges of the same magnitude: + Q and – Q, placed at constant distance l the electric dipole moment direction from – Q to +Q 𝑝=𝑄ℓ pro 𝑟≪ℓ     Q1 Q2 │Q1 │ = │ Q2 │ Physics I-2019, Lecture 7

iii) Electric field for continuously distributed charge: for charge distributed over a long wire ℓ : the linear charge density for charge distributed over a plane 𝑆: the surface charge density for charge distributed over a certain volume 𝑉: the volume charge density 𝑄= ℓ 𝜆𝑑ℓ 𝜆= lim Δℓ→0 ∆𝑄 Δℓ = 𝑑𝑄 𝑑ℓ 𝑄= 𝑆 𝜎𝑑𝑆 𝜎= lim Δ𝑆→0 ∆𝑄 Δ𝑆 = 𝑑𝑄 𝑑𝑆 𝑄= 𝑉 𝜌𝑑𝑆 𝜌= lim Δ𝑉→0 ∆𝑄 Δ𝑉 = 𝑑𝑄 𝑑𝑉 Physics I-2019, Lecture 7

 s <0 s >0 Homogeneous electrostatic field 14-6, ex. 2 𝐸 = 𝐹 𝑄 0 Homogeneous electrostatic field 14-6, ex. 2 = the field vector 𝐸 is in this region constant filed of infinite sheet of charge, distributed uniformly with a surface charge density s [C/m2], estimation: Two parallel plates with a charge density +s and – s, in distance d s <0 s >0 𝐸 (−) = 𝜎 2 𝜀 0  𝐸=0 𝐸= 𝜎 𝜀 0 𝐸=0 between indefinite plates – homogeneous field, outside zero

the work of an el. force in el. field 𝐸 to move a charge Q from 𝐴→𝐵 𝐹 =𝑄 𝐸 Work and potential the work of an el. force in el. field 𝐸 to move a charge Q from 𝐴→𝐵 El. field is conservative, we can introduce potential energy potential V – potential energy of an unit charge potential difference UAB (voltage) – difference between potentials 𝑊 𝐴→𝐵 =𝑄 𝐴 𝐵 𝐸 ∙𝑑 𝑟 𝐸 𝑝 𝑟 =𝑄 𝑟 𝐸 𝑝 =0 𝐸 ∙𝑑 𝑟 𝑉( 𝑟 )= 𝑟 𝑉=0 𝐸 ∙𝑑 𝑟 unit: V (volt) unit of el. intensity: Vm-1 scalar quantity describing el. field 𝑉( 𝑟 )= 𝐸 𝑝 ( 𝑟 ) 𝑄 𝑈 𝐴𝐵 = 𝑉 𝐴 − 𝑉 𝐵 𝑈 𝐴𝐵 = 𝐴 𝐵 𝐸 ∙𝑑 𝑟   Physics I-2019, Lecture 7

𝑉( 𝑟 )= 𝑟 𝑉=0 𝐸 ∙𝑑 𝑟 board 𝑉 𝑟 =𝑘 𝑄 𝑟 𝑉= 𝑖=1 𝑛 𝑉 𝑖 = 𝑖=1 𝑛 𝑘 𝑄 𝑖 𝑟 𝑖 𝑉( 𝑟 )= 𝑟 𝑉=0 𝐸 ∙𝑑 𝑟 i) potential of the field of a point charge board ii) potential of a group of point charges 𝑉 𝑟 =𝑘 𝑄 𝑟 V > 0 for Q > 0 to move a Q > 0 to infinity – the field perform a positive work V < 0 for Q < 0 to move a Q > 0 to infinity, positive work of external force, the field performs a negative work el. potential is a scalar, indirectly proportional do the distace not defined in a point charge, i.e. for r = 0 equipotential surface ≡ surface of constant potential 𝐸  equipotential surface (generally valid) pot. energie a group of point charges 𝑉= 𝑖=1 𝑛 𝑉 𝑖 = 𝑖=1 𝑛 𝑘 𝑄 𝑖 𝑟 𝑖 𝐸 𝑝 = 𝑑𝑣𝑜𝑗𝑖𝑐𝑒 𝑘 𝑄 𝑖 𝑄 𝑗 𝑟 𝑖𝑗 Physics I-2019, Lecture 7

iii) voltage between two sheets, +s a – s, distance d board 𝑈 𝐴𝐵 = 𝐴 𝐵 𝐸 ∙𝑑 𝑟 iii) voltage between two sheets, +s a – s, distance d board 𝑊 𝐴𝐵 =𝑄 𝐴 𝐵 𝐸 ∙𝑑 𝑟 𝑈=𝐸𝑑 the work to move a charge 𝑄 from one sheet do the second one: d 𝑊=𝑄𝑈=𝑄𝐸𝑑 x Physics I-2019, Lecture 7

Motion of a charged particle in an e. field linear accelerator a charge Q of mass m enters hom. field wit a velocity 𝑣 0 parallel to the field 𝐸 velocity after passing a voltage U, board for 𝑣0 = 0 𝑣= 2𝑄𝑈 𝑚 = 2𝑄𝐸𝑑 𝑚 Physics I-2019, Lecture 7

El. dipole in a homogeneous el. filed Goal: state of motion of a dipole of el. dipole momentum p net force → no translation only rotation: 𝐹 =𝑄 𝐸 𝐹 = 0 momentum of force board potential energy 𝑀 = 𝑝 × 𝐸 𝐸 𝑝 (𝛼)=− 𝑝 ∙ 𝐸 𝐸 𝑝 (𝛼)=−𝑝𝐸 cos 𝛼 Physics I-2019, Lecture 7

El. dipole in a homogeneous el. filed important position of a dipole in hom. field el. dipole tends to rotate into stabile equilibria 𝐸 𝑝 (𝛼)=− 𝑝 ∙ 𝐸 𝑀 = 𝑝 × 𝐸 𝐸 𝑝 𝛼 =−𝑝𝐸 cos 𝛼 𝑀=𝑝𝐸 sin 𝛼 25 Physics I-2019, Lecture 7

7.5 Electric field in medium Physics I-2019, Lecture 7