Unit 2: Electric properties of conductors and dielectrics. Charged conductors in electrostatic equilibrium. Ground. Electrostatic influence. Electric shield. The parallel-plate capacitor. Capacitance. Stored energy in a capacitor. Combination of capacitors. Electric dipole. Dielectrics. Capacitors with dielectric
Charged conductors in electrostatic equilibrium Conductors: Materials whose electric charge (electrons) can move from any point to other due to an electric field. By adding e - Net charge – By removing e - Net charge + Dielectrics: The electrons are firmly linked to atoms and net charge can not change. Dielectrics can only be polarized. Tipler, chapter 22, section 22.5
Charged conductors in electrostatic equilibrium Conductors in electrostatic equilibrium: There isn’t net movement of the charges (F=0). As electric forces are due to an electric field: � � � F qE 0 E 0 = = = Electric field inside a conductor in electrostatic equilibrium is zero at any point of the conductor. Tipler, chapter 22, section 22.5
Charged conductors in electrostatic equilibrium Electric charge in a conductor must reside on the conductor’s surface. conductor’s inside d S E = 0 φ = E ⋅ = 0 S Gauss’s Gauss’s surface (S) theorem Q i φ = E ε 0 Q = 0 i Electric charge must reside on conductor’s surface
Charged conductors in electrostatic equilibrium Any conductor’s point has equal electric potential: 0 B E = V V d l 0 V V − = − ⋅ = B A A B A B A Tipler, chapter 23, section 23.5
Charged conductors in electrostatic equilibrium Electric field near the conductor’s surface is perpendicular to conductor’s surface. If electric field wasn’t perpendicular, the tangential component E t should move the charges and so the conductor wouldn’t be in equilibrium. E E E n E t F = qE t Charge moving Charge not moving
Charged conductors in electrostatic equilibrium Coulomb’s theorem: at points near conductor’s surface σ E = ε S 0 � E u It can be demonstrated by applying Gauss’s law
Charged conductors in electrostatic equilibrium Summary of properties of charged conductors in electrostatic equilibrium: E=0 inside the conductor. All the charge must be on the surface as σ . There isn’t charge inside the conductor. Electric potential is constant in all the conductor V=cte. Electric field near the conductor’s surface is perpendicular to the surface, with a value: E s = σ / ε 0
Hollow conductor The behaviour of a hollow conductor without charges inside is the same as solid conductor: q � � E = 0 E = 0 V cte = V cte = 0 σ = i
Q R R r r q 2 Sharp tip effect Q 1 Exercise 2.2 V V q Q q Q + = 1 2 QR Qr Q Solution: Q q 2 = V k R 1 = = Q q R r R r + + ( r ) + 1 2 V k k = = R r E 2 > E 1 Q QR Q V σ 1 1 E σ = = = = = Sharp tif 1 1 The lower the radius the higher the 2 2 4 R ( R r )4 R 4 ( R r R ) R π + π ε πε + 0 0 electric field near the conductor. effect: q Qr Q V σ 2 2 Sharp tips attract the electric charges E σ = = = = = 2 2 2 2 4 r ( R r )4 r 4 ( R r r ) r π + π ε πε + 0 0 Sharp tip effect: due to the high electric field near a sharp tip: - Lightning rod - An umbrella during a storm - St. Elmo’s fire (fuego de San Telmo) - http s://www. youtube .com/watch?v= kdNjKdmpkOs
Ground Electric potential of a spheric Q V = conductor is given by: 4 πε R 0 As Earth has a very big radius (R →∞ ) related to any object, electric potential of earth (ground) is zero for any charge Q. Ground can take or give any charge without change its electric potential (it’s like the sea level) V = 0 G Connecting a device to ground means safety for people
Linking a conductor to Ground Linking a conductor to Ground ( ) means: 1. Electric potential is 0 (V=0) 2. The conductor can change its charge by taking or giving electrons to Ground. � E = 0 Without charges inside V = 0 E=0 q V=0
Electrostatic influence When we put an electric charge near a conductor, electrostatic influence divides the charge E = 0 inside the conductor. i E E = 0 i
Total electrostatic influence Total Electrostatic influence between two conductors occurs when all the field lines starting from a conductor end in the other conductor. Surfaces with total influence have the same charge but different sign -Q +Q +Q -Q
Electric shield or Faraday’s cage A hollow conductor linked to ground divides electrically the inner and outer spaces. It’s known as an electric shield. Outer charges don’t influence inner space……… � E q � E = 0 V = 0 σ e 0 σ = i
Electric shield or Faraday’s cage And inner charges don’t influence outer space. � E = 0 V = 0 � E q σ i σ = 0 e
The parallel-plate capacitor It’s made up by two parallel plate conductors being its surface much more greater than the distance between them (Total electrostatic influence). Tipler, chapter 24, section 24.1
The parallel-plate capacitor. Capacitance S If a parallel-plate capacitor is charged with a charge Q (+Q on a plate and –Q on the + σ other plate) (in vacuum): E -Q +Q σ Q E = - σ σ = ε S 0 and the difference of potential between the plates: d � − � σ d + − V = V − V = E ⋅ d r = E ⋅ d = ε 0 +
The parallel-plate capacitor. Capacitance The rate Q/V is known as the capacitance (C) of the capacitor, and it’s depending on the geometry (size, shape and relative position), and not depending of the charge of the capacitor: Q σ S ε S 0 = = ε = C 0 V σ d d [C]=M -1 L -2 T 4 I 2 Unit: Farad (F) C
Some parallel-plates capacitors
Other capacitors. Cilindric capacitor π ε 2 L 0 C = ( ) ln r / r 2 1
Combination of capacitors. Capacitors in series When many capacitors are connected in series, all the capacitors have the same charge. 1 1 1 1 1 = ... = + + + C C C C C Tipler, chapter 24, section 24.3 i eq 1 2 3 i
Combination of capacitors. Capacitors in parallel When many capacitors are connected in parallel, all the capacitors have the same difference of potential. = C C C C ... C = + + + eq 1 2 3 i Tipler, chapter 24, section 24.3 i
Stored energy in a capacitor To charge a capacitor means to carry charge from a plate to another plate (negative charge from + to -, or positive charge from – to +) . Let us take the situation where the charge and the potential of capacitor are q and V. To increase a dq charge, must be done a work (dU): q q - v = dU vdq dq = = C C Tipler, chapter 24, section 24.2
Stored energy in a capacitor To charge a discharged capacitor until Q charge, the work done (stored as energy on the electric field) will be: Q Q 2 q Q 1 U dU vdq dq = = = = C C 2 0 0 From capacitance definition: 2 1 Q 1 1 2 U = = QV = CV 2 C 2 2 Tipler, chapter 24, section 24.2
Dielectrics. Dipolar polarization. Dielectrics have no free electrons. But their polar molecules (dipoles) can be oriented by an electric field (dipolar polarization). They are randomly oriented when Polar molecule no electric field is acting. water F=qE � E 0 Tipler, chapter 24, sections 24.5 and 24.4
Dielectrics. Ionic polarization. It occurs on dielectrics with non polar molecules. When an electric field acts, molecules become polars, they turn and polarization occurs (ionic polarization). F=qE 0 Acting an external electric field, centers of positive and negative charge are displaced, � resulting on electric dipoles. E 0 Dipoles are oriented when a electric field is acting Tipler, chaper 24, sections 24.5 and 24.4
Dielectrics. Behaviour on an electric field. Whatever the type of polarization(dipolar or ionic), an opposite (E d ) to the original electric field (E 0 ) appears. The resulting electric field E is lower than the original. E d E=E o -E d =E 0 / ε r < E o ε r (or k ) is characteristic for each material, and it’s called relative dielectric permitivity or dielectric constant. E 0 ε r ≡ k goes from 1 to ∞
Capacitor with dielectric. Let’s take an isolated capacitor with some charge Q Q σ E Q E = = 0 E = ε = 0 S ε ε S ε ε 0 0 r 0 r Qd V E d Qd V = = Q Q 0 0 0 V Ed = = = S -Q ε 0 S ε ε ε 0 r r Q S ε Q S ε ε 0 C V = = V 0 r C C C = = = ε > 0 V d r 0 0 V d 0 Isolated Capacitor without Isolated Capacitor with dielectric dielectric The effect to fill a capacitor with a dielectric is the increasing on capacitance. It is multipied by the relative dielectric constant: C C C = ε > r 0 0 Tipler, chapter 24, section 24.4
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