1. 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

2. 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

3. 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

4.  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

5. 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

6. 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

7. 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

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

9. 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

10. 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

11. 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

12. - 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, 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

13. 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 𝐸 𝑃
er – relative permittivity (no units)
er (vacuum)= 1
𝐸= 𝐸 0 − 𝐸 𝑃 = 𝐸 0 𝜀 𝑟
Physics I-2019, Lecture 8

14. 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

15. 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= 𝑄 2 𝐶 = 1 2 𝐶 𝑈 2
𝑤= 1 2 𝜀 𝑟 𝜀 0 𝐸 2
Physics I-2019, Lecture 8

16. 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

17. 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

18. 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

19. 8.3 Electromotive force and current circuits
Physics I-2019, Lecture 8