7.3 Electric field in vacuum 7. Electrostatic field … 7.3 Electric field in vacuum 7.4 Motion of a charged particle in an electric field 7.5 Electric field in medium Direct current circuits 8.1 Electric current 8.2 Ohm’s law 8.3 Electromotive force and current circuits Physics I-2019, Lecture 8

the field of a point charge Electric field: 14-9 Deff. the field of a point charge The field of a dipole 𝐸 = 𝐹 𝑄 0 Physics I-2019, Lecture 8

iii) continuously distributed charge, quantities: 14-2, p.113 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 𝜌𝑑𝑉 𝜌= lim Δ𝑉→0 ∆𝑄 Δ𝑉 = 𝑑𝑄 𝑑𝑉 Physics I-2019, Lecture 8

 s <0 s >0 Homogeneous electrostatic field 14-6, example 2 𝐸 = 𝐹 𝑄 0 Homogeneous electrostatic field 14-6, example 2 = the field vector 𝐸 is in this region constant field 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 infinite plates – homogeneous field, outside zero

the work by an el. force in el. field 𝐸 to move a charge Q from 𝐴→𝐵 𝐹 =𝑄 𝐸 Work and potential 14-8 the work by 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. field: 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 8

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 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 to the distance not defined in a point charge, i.e. for r = 0 equipotential surface ≡ surface of constant potential 𝐸  equipotential surface (generally valid) 𝑉= 𝑖=1 𝑛 𝑉 𝑖 = 𝑖=1 𝑛 𝑘 𝑄 𝑖 𝑟 𝑖 Physics I-2019, Lecture 8

iii) voltage between two infinite sheets, +s a – s, distance d board 𝑈 𝐴𝐵 = 𝐴 𝐵 𝐸 ∙𝑑 𝑟 iii) voltage between two infinite 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 8

Motion of a charged particle in an el. field Example: linear accelerator a charge Q of mass m enters hom. field with a velocity 𝑣 0 parallel to the field 𝐸 determine the velocity after passing a voltage U, board for 𝑣0 = 0 𝑣= 2𝑄𝑈 𝑚 = 2𝑄𝐸𝑑 𝑚 Physics I-2019, Lecture 8

El. dipole in a homogeneous el. filed 14-4 (p. 116) 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 forces board potential energy 𝑀 = 𝑝 × 𝐸 𝐸 𝑝 (𝛼)=− 𝑝 ∙ 𝐸 𝑀=𝑝𝐸 sin 𝛼 𝐸 𝑝 (𝛼)=−𝑝𝐸 cos 𝛼 Physics I-2019, Lecture 8

El. dipole in a homogeneous el. filed important position of a dipole in hom. field el. dipole tends to rotate into stable equilibria 𝐸 𝑝 (𝛼)=− 𝑝 ∙ 𝐸 𝑀 = 𝑝 × 𝐸 𝐸 𝑝 𝛼 =−𝑝𝐸 cos 𝛼 𝑀=𝑝𝐸 sin 𝛼 stable equilibria Physics I-2019, Lecture 8

7.5 Electric field in medium 14-7, 14-12 conductors – some of charged particles can move “rather” freely statement: In static situation , the electric field inside 𝐸 = 0 . proof by contradiction consequence: The charge on conductor distributes itself on the outer surface. statement: The direction of 𝐸 close to surface is perpendicular to the surface. proof by contradiction insulators, dielectric – not vacuum and not conductor general direction - tangential component exists - motion of charges = contradiction → Physics I-2019, Lecture 8

- p(molecule) = 0 p(molecule) ≠0 Dielectrics dielektrics polar nonpolar p(molecule) = 0 nonpolar molecules when E ≠ 0 Vm-1, p(molecule) ≠ 0 polar effective center of + and – do not coincide - p(molecule) ≠0 polar molecules: p~10-30 Cm unit used in chemistry (debye): 1D=3,336. 10-30 Cm + - + - + - without el. field any volume of dielectric is nonpolar (dipoles of molecules of polar dielectrics - thermal agitations) in external filed - polarization in nonpolar dielectrics – a slight net displacement of the effective centers of charge, the dipole is induced in polar dielectrics - 𝑝 aligns parallel to field (not perfectly, against it - thermal agitations) general description of the both cases the same Physics I-2019, Lecture 8

er – relative permittivity (no units) polarization of dielectrics - a slab of dielectrics induced surface charges – bound in dielectrics, cannot move freely free charge – in conductor the net field in dielectrics – superposition of the field of the free and the surface charge: permittivity of medium 𝜀=𝜀 𝑟 𝜀 0 relations in vacuum → relations in dielectrics: e0 → e examples board 𝐸 = 𝐸 0 + 𝐸 𝑃 er – relative permittivity (no units) er (vacuum)= 1 𝐸= 𝐸 0 − 𝐸 𝑃 = 𝐸 0 𝜀 𝑟 Physics I-2019, Lecture 8

S… area of each plate, s… surface charge density, Capacitor 14-10 two conductors with charges +Q and –Q , the voltage between them is U Def. capacitance parallel-plate capacitors: ~ S ~ 1/d ~ er capacitance of a vacuum capacitor: capacitance of capacitor with dielectrics C = er C0 combination of capacitor in series in parallel 𝐶= 𝑄 𝑈 unit: F (farad) constant for given capacitor S… area of each plate, s… surface charge density, d…distance between the plates, charge on plates +Q, -Q board 𝐶= 𝜀 0 𝜀 𝑟 𝑆 𝑑 𝐶 0 = 𝜀 0 𝑆 𝑑 Physics I-2019, Lecture 8

Energy of electrostatic field 14-11, p. 130, 131 𝐶= 𝑄 𝑈 Energy of electrostatic field 14-11, p. 130, 131 simplified: energy of the el. field in parallel-plate capacitor energy ≡ work done to charge it on voltage 𝑈 board energy density w = energy per unit volume valid generally in the field E 𝑊= 1 2 𝑄 2 𝐶 = 1 2 𝐶 𝑈 2 𝑤= 1 2 𝜀 𝑟 𝜀 0 𝐸 2 Physics I-2019, Lecture 8

8. Direct current circuits 15 8.1 Electric current 15-1 electrodynamics electric current – definition of process directed motion of electric charges (conductors in an electric field) Def. of quantity: i (t), I Def: current density – vector which is characteristic of a point in conductor rather than a conductor a net charge that passes trough a cross section of conductor in the time unit unit of electric current A (ampere) unit of charge C=A s positive direction of current ≡ direction of movement of positive charges 𝐼= lim ∆𝑡→0 ∆𝑄 ∆𝑡 = 𝑑𝑄 𝑑𝑡 direction that positive charge carrier would move in the point 𝑑𝐼= 𝑗 ∙𝑑 𝑆 𝐼= 𝑆 𝑗 ∙𝑑 𝑆 el. current is a flow of the current density vector thorough cross section 𝑆 Physics I-2019, Lecture 8

interpretation in a simple case: unit of current density: A m-2 𝐼= 𝑆 𝑗 ∙𝑑 𝑆 el. current is a flow of the current density vector thorough 𝑗= 𝐼 𝑆 𝐼= 𝑑𝑄 𝑑𝑡 Physics I-2019, Lecture 8

current trough metal conductors is proportional to the applied voltage 8.2 Ohm’s law 15-2 relation between current and voltage, originally for metal conductors 𝐼= 𝑈 𝑅 current trough metal conductors is proportional to the applied voltage 𝑅 … resistance of wire, unit Ω (ohm) = V/A 𝑈=𝑅𝐼 Ohm’s law not valid for semiconductors, transistors, vacuum tubes, etc. so called nonohmic materials resistance for a uniform metal wire of lengths ℓ and cross section 𝑆: Resistance dependence on temperature (metal wires) 𝑅= 1 𝜎 ℓ 𝑆 =𝜚 ℓ 𝑆 𝜎 … conductivity, 𝜎= 1 𝜚 𝜚 … resistivity, [r] = m, depends on material 𝜚 𝑇 … resistivity at temperature 𝑇 𝜚 0 … known resistivity at standard temperature 𝑇 0 𝛼 … the temperature coefficient of resistivity; for metals 𝛼>0, for semiconductors 𝛼<0 𝛼 for semiconductors can be <0 𝜌 𝑇 = 𝜌 0 1+𝛼 𝑇− 𝑇 0 Physics I-2019, Lecture 8

8.3 Electromotive force and current circuits Physics I-2019, Lecture 8