ELECTRIC POTENTIAL ENERGY

ELECTRIC POTENTIAL ENERGY Before considering the electric potential energy, let’s remind what we studied last year about the gravitational field:

Gravitational potential energy A body which is at a certain level from the ground possesses, with reference to the ground, a gravitational potential energy. This energy is equivalent to the work that the weight of the body (body weight) would perform in the fall; and it is then calculated by: U = m∙g∙h Where: m = mass g = the acceleration of gravity h = height U = gravitational potential energy Gravitational Potential Energy is the energy a body has got due to its elevation (height) with reference to a lower elevation, that is, the energy that could be obtained by letting it falls to a lower elevation.

The potential energy, therefore, represents the energy stored by a body that is the work that a body could fulfill if it had moved, but in reality, in that moment, it didn’t do. Example : What is the potential energy of a body of 4 kg arranged at a height of five metres from the ground? m = 4 Kg h = 5 m Ep = ? U = m∙g∙h = 4 ∙ 5 ∙9,8 = 196 J This is the work that the body will do in the fall. This is also the work that has been done to bring the body from the horizontal reference to height (h); the work of the body will return as previously said during the fall

Work and potential energy Suppose a body with a mass m carries out a distance from point A to point B ( In this case, we indicate h = h1 – h2 , the work is indicated: W = mgh = mg(h1 – h2) = mgh1 – mgh2 W = mgh1 – mgh2 If: U= mgh U1=mgh1 and U2=mgh2 W = mgh1 – mgh2 = U1 – U2 that is The gravitational force fulfills a work which is the result considering the variation of the potentian energy but changing its positive mark into a negative one. If the work done is positive, the final potential energy decreases, what it happens when a body falls down from a given height to the ground; if , instead, it is negative, the final potetial energy encreases , what it happens when we lift a body from the ground to a given height. N.B. Generally speaking: a force is conservative, like a weight force, when the work doesn’t depend on the covered route from A to B but it depends only on the position of the two extreme points.

The quantity mgh represents the gravitational potential energy. It depends from the height h where the body is situated, and from the mass m of it. If we divide the potential energy by the mass of the body, we obtain a new physical quantity, called potential: It is called potential (indicated with the symbol V), the potential energy divided by the mass.. Let’s consider again the work formula : W = mgh1 – mgh2 that we can also write in this way: Considering the differece: It is positive : W > 0 It means that during the body fall (to go from point A to point B), the work done by the gravitational field is positive. If the work is positive itmeans that, during the fall, the work is done by the force of the field, and so we have that the movement is spontaneous. All the bodies in a gravitational field move spontaneously from points which have greater potential energy ( and then potential) to points which have a lower potential energy. Instead, to go from point B to point A,, the work is negative, it means that during the lifting (raising) of the body the work is done by an outward force to the gravitational field that, which so it is opposite to the body motion.

Electric & Gravitational - Fields Compared Gravity: Electric Force: Field strength is given by per unit mass or force per unit charge, depending on the type of field. Field strength means the magnitude of a field vector. Ex #1: If a +10 C charge is placed in an electric field and experiences a 50 N force, the field strength at the location of the charge is 5 N/C. The electric field vector is given by: E = 5 N/C, where the direction of this vector is parallel to the force vector (and the field lines). Ex #2: If a -10 C charge experiences a 50 N force, E = 5 N/C in a direction opposite the force vector (opposite the direction of the field lines).

Electrical potential energy W = Fh = F(h1 – h2) = qE(h1 – h2) W = qEh1 – qEh2 = q(Eh1 – Eh2) W = q(V1 – V2) There is a positive charge q inside the uniform electric field, produced (generated) by two conductor parallel plates; S’ charges positively and S charges negatively. When the charge moves parallel to the field E, to go from point A to point B, we have: where the quantity: qEh1indicates the electrical potential energy of the charge q in the point A: U1= qEh1 W = qEh1 – qEh2 It is called potential (indicated with V, as symbol) the potential energy divided by the charge : So there is a relation between the potential and the electric field : The measure unit : Therefore we have : W = qEh1 – qEh2=q(Eh1- Eh2) = q(V1 – V2) and so also W > 0 As the potential energy in A is greater than in B, the quantity : qEh1 – qEh2 > 0 It means (W > o ) that a positive charge moves spontaneously, under the action of the electric field, from points with a greater potential energy to points with a smaller (lower) potential energy).

Examples 1-How much is the work done by the field forces to move a positive charge of q=2μ C from a point with a 200 V potential to another point with a 100 V potential? W = q(V1 – V2) = 2μ (200 – 100) = 2 10-4 J 2- And if the charge would be negative, how much would the work be ? 3-How much is the work done by the field forces to move a positive charge of q=2 μ C from a point with a 100 V potential to another point with a 200 V potential 4- To move a charge of q=2 μ C from a point A to another point B, the forces of the field do a work of 10J. What is the difference of potential between the two points?

Potential in a point of a field generated by a point charge. Let’s calculate the potential produced by a charge Q in the point P, far d from the charge Q: It means that increasing the distance the potential decreases.. But then : Example 1- Calculate the potential produced by a charge Q= 4*10-6 C at a distance of 10 cm from the charge. 2- The potential, at a distance of 5 m from the charge, is 20 V. How much is the electric field in this point ?

ELECTROSTATIC EQUILIBRIUM CONDITIONS We have considered a conductor in electrostatic equilibrium where the charge is spread in all its surface, so that inside of the conductor the electric force is =0. We can say so: E = 0 inside the conductor V=cost. So, we have: In an equivalent way, we have: E=0 That is: inside the conductor What it is said above expresses the conditions of the electrostatic equilibrium in the conductors

Equipotential Surfaces We define equipotential surfaces those ones where the potential is constant. Let’s calculate the work to move a charge q from point A to B along an equipotential surface. We know that: : as in any equipotential surface V = cost, it will be: VA = VB, so: VA - VB = 0 W = q(VA – VB) Therefore: W= 0 that is: work is zero As both the field force, that does the work (electric force) , and the displacement are not zero, as we know because the work results zero the condition is that the force has to be perpendicular to the displacement. But the displacement can be anywhere over the equipotential surface: then the force is perpendicular to the equipotential surface and if the force is perpendicular so the field lines are also perpendicular. Conclusion: The equipotential surfaces have the property to be perpendicular to the field lines.

a) If we have a charged plane plate, the equipotential surfaces are plane parallel to the plate and the field lines are perpendicular to the equipotential surfaces b) In case we have (Considering) a point charge, the equipotential surfaces are concentric spheres to the charge that produces the field and the field lines are perpendicular to the equipotential surfaces

Capacitance The electric field, inside a charged conductor in electric equilibrium, is zero; if it weren’t there would be a motion of charges and then the initial condition of electric equilibrium of conductor wouldn’t be fulfilled. In the same way the potential inside the conductor is constant and it has the same value that the surface has got: that is all the points of the conductor have the same potential, its value is: indicating with r the radius of a spherical conductor. The potential of a conductor is proportional to the charge it has got, so: The quantity C is called electric capacitance of the conductor. That is: The capacitance is measured in farad: As the farad is a very great unit, we often use the submultiples: microfarad (1μF = 10-6F ; nanofarad (nF = 10-9F) ed il picofarad (pF = 10-12F) The capacitance of a spherical conductor is : dielectric constant in the empty space

Capacity of a spheric conductor We have seen the capacity of a spheric conductor is: Now, if C=1 F, the value of the radius has to concide with the value of the numerical constant that in the vacuum is: 9*109 That means a spheric conductor in the vacuum, in order to have the capavity of one farad, should have a radius of 9*109 m that is 9 million of km. Reminding the earthly radius is an average of 7,000 km then we couldn’t consider as conductor the Earth either in order to have a capacity of 1F. The Farad is such a great and huge measure unit and for this reason we use its submultiples. Exercises 1- What is the capacity of a spherical conductor with a radius of 2 mm? 2- What is the capacity of the Earth (radius = 7,000 km)?

CAPACITORS • We can verify that the capacitanceof a conductor increases even though we place nearby a discharged conductor. • Capacitors (formed with two conductors) are devices which are capable to have a very great electric capacitance in small dimensions. • According to the shape of the conductors, we have different types of capacitors: • - plate capacitor; • - spherical capacitor; • cylindrical capacitor. There are different types of capacitors on sale. Let’ consider a capacitor made up by two conductors with two flat and parallel plates. (a) (b) Graphic symbol of a capacitor Figure (a) shows a parallel plate capacitor connected to a battery, and (b) is the corresponding circuit Flat metal plates ofa capacitor

Capacitors A capacitor consists of two conductors (with area A) that are close together, but not touching (at distance d). A capacitor has the ability to store electric charge. • A capacitor is a device that stores electrical charge. • A charged capacitor is actually neutral overall, but it maintains a charge separation. • The charge storing capacity of a capacitor is called its capacitance. • An electric field exists inside a charged capacitor, between the positive and negative charge separation. • A charged capacitor store electrical potential energy • in the electric field. • Capacitors are ubiquitous in electrical devices. They’re used in power transmission, computer memory, photoflash units in cameras, tuners for radios and TV’s, defibrillators, etc. Where the constant K depends on the capacitor and it is indicated by the symbol C, said capacitance. When a capacitor is connected to a battery, the charge on its plates is proportional to the voltage: q = K V

Parallel Plate Capacitor CHARGE OF A CAPACITOR The simplest type of capacitor is a parallel plate capacitor, which consists of two parallel metal plates, each of area A, separated by a distance d. When one plate is attached via a wire to the + terminal of a battery, and the other plate is connected to the - terminal, the battery pulls e-’s from the plate connected to the + terminal (so this plate charged +) and deposits them on the other (charged - ). -Q +Q Area, S d V wire Symbols used in the circuit diagram to the left The energy stored in a capacitor lies in the electric field between the two plates. In a capacitor with two flat metal and parallel plates the electric field between the two plates is uniform.

DISCHARGE OF A CAPACITOR A charged capacitor discharges quickly if the two plates are connected with a conductor wire, or if there is between them any type of contact so to let electrons passed on. This transfer of energy during the discharge is considerable and sudden.

Capacitance Capacitance, C, is the capacity to store charge. The amount of charge, Q, stored on given capacitor depends on the potential difference between its plates, V, and its capacitance C. In other words, Q is directly proportional to V, and the constant of proportionality is C. The capacitance of a parallel plate capacitor filled with a dielectric is: A dielectric is an insulator, and it is put between the two plates. It is characterised by a dielectric constant It is quite obvious that different substances have different dielectric constants in case the dielectric is any substance In case the dielectric is the empty space (vacuum): Vacuum dielectric costant is

Capacitance: SI Units The SI unit for capacitance is the farad, named for the famous 19th century scientist Michael Faraday. Its symbol is F. From the defining equation for capacitance, Q = CV, we define a farad Q = C V implies 1 C = (1 F) (1 V) So, a farad is a coulomb per volt. This means a capacitor with a capacitance of 3 F could store 30 C of charge if connected to a 10 V battery. This is a tremendous amount of charge for a reasonable potential difference. Thus a farad is a large amount of capacitance. Many capacitors have capacitances measured in pF or fF (pico or femtofarads). m: milli= μ: micro = n: nano = p: pico = f: femto =

For a parallel plate capacitor as shown, the field between the plates is: The potential difference between the plates is: The capacitance of a parallel plate capacitor is therefore:

Capacitance Problem 1 A parallel plate capacitor is fully charged by a 20 V battery, acquiring a charge of 1.62 nC. The area of each plate is 3.5 cm2 and the gap between them is 1.3 mm. What is the capacitance of the capacitor? From Q = C V, C = Q / V = (1.62  10-9 C) / (20 V) = 8.1  10-11 F = 81  10-12 F = 81 pF. The gap and area are extraneous. -1.62 μC 3.5 cm2 1.3 mm 20 V

Applicazioni dei condensatori Il fatto che C dipenda da d viene utilizzato per costruire le tastiere del computer: il tasto è appoggiato sopra un condensatore che ha una determinata capacità. Quando si schiaccia il tasto, vedi figura, d diminuisce e quindi la capacità del condensatore aumenta. I circuiti elettronici della tastiera rilevano la variazione di capacità e segnalano al computer che il tasto è premuto. Quando tocchiamo lo schermo touch screen di un dispositivo con un dito andiamo ad aggiungere un conduttore che prima non c’era, e ciò fa cambiare le proprietà capacitive di quella zona; cioè modifichiamo il campo elettrico all’interno di un condensatore di cui l’intero schermo costituisce una delle due armature. Si crea così un segnale che viene rilevato ed interpretato dal software del dispositivo Infine, quando giriamo una manopola di una radio, per sintonizzarci su di una data frequenza, stiamo utilizzando un condensatore a capacitò variabile; in questo caso varia la superficie delle armature affacciate.

Heart defibrillators use an electrical discharge to “jump-start” the heart, and therefore to save lives. Flash lamps require a very high current over a very short time period to operate, photoflash capacitors are designed for this purpose. Touch screen panels makes use of capacitors. They consists of an insulator, glass for example, coated with a transparent conductor. The human body is a good electrical conductor and when the surface of the screen is touched by the fingers there is a variation in the screen’s capacitance, which is detected.

Capacitance Problem 2 A parallel plate capacitor has the area of its plates S= 0,060 m2. We know that the distance of the plates is d=1,4 mm. The plates are charged Q = 1,9x10-9 C . What is the capacitance of the capacitor? What is the potential difference between the plates and the electric field?

Exercises 1- Between the plates of a capacitorthereis a d.d.p. of 50 V. If the charge of the platesis 1*10-6 C, howmuchis the capacitance of the capacitor? 2- The plates of a capacitor of 10μF capacitance have a charge of 0.25 C, What is the potential difference of the plates? 3- A capacitor is formed by two plane and parallel plates with an area of S= 0.04 m2, distant 0.2 cm, empty inside. Value the electric charge in the plates when the capacitor has a potential difference of 220 V .

Exercise A capacitor is formed by two plane and parallel plates with an area of 0.034 m2; they are distant 0.15 cm, empty inside. The plates have a charge of 3*10-9 C. - What is the capacitance of the capacitor? - How much is the uniform electric field inside the plates of the capacitor? How can the capacitors be connected to each other ? When using the connection in series and when, instead, the one in parallel ?

Capacitors in Parallel Systems including capacitors more than one has equivalent capacitance. Capacitors can be connected to each other in two ways. They can be connected in series and in parallel. We will see capacitors in parallel first. Because, left hand sides of the capacitors are connected to the potential a, and right hand sides of the capacitors are connected to the potential b. In other words we can say that each capacitor has same potential difference and so V is constant. We find the charge of each capacitor Q1=C1*V Q2=C2*V Q3=C3*V Total charge of the system is found by adding up each charge: Qtotal= Q1+Q2+Q3 = C1*V+C2*V+C3*V = V*(C1+C2+C3) but: Q=C*V →C*V = V*(C1+C2+C3)→ Ceq=C1+C2+C3 As you can see, we found the equivalent capacitance of the system as C1+C2+C3 N.B. Parallel connection is useful when high capacity systems are required.

Capacitors in series In capacitors in series, each capacitor has same charge flow from battery. In this circuit, +Q charge flows from the positive part of the battery to the left plate of the first capacitor and it attracts –Q charge on the right plate, with the same idea, -Q charge flows from the battery to the right plate of the third capacitor and it attracts +Q on the left plate. Other capacitors are also charged with same way. To sum up we can say that each capacitor has same charge with batter and so Q is constant. C1*V1=Q C2*V2=Q C3*V3=Q V=V1+V2+V3= And Ceq.*V=Q N.B. The serial connection is useful when you need to split a certain d.d.p. Without altering its value.

Capacitors in parallel have the same voltage across each individual capacitor. The equivalent capacitor is one that stores the same charge when connected to the same battery: Capacitors in series have the same charge. In this case, the equivalent capacitor has the same charge for the same voltage difference. The inverse of the equivalent capacitance is therefore given by the sum of the inverses of the individual capacitances:

Esercizio: Tre condensatori hanno rispettivamente capacità 4mF, 8mF e 1 2mF. Essi vengono collegati prima in serie e poi in parallelo. Calcolare la capacità del sistema nei due casi. Parallelo C = C1 + C2 + C3 = 4 + 8 + 12 = 24mF Serie:

The energy employed to charge a capacitor, and hence the energy stored in the capacitor, is given by Energia Per caricare un condensatore si esegue un dato lavoro: Se indichiamo con V la d.d.p. ai capi del condensatore e con Q = C*V la carica positiva che si ha su una delle due armature, alla fine del processo di carica il lavoro è dato da : oppure Energy stored is equal to the work done to charge the capacitor. oppure Per la conservazione dell’energia, il lavoro compiuto per caricare il condensatore rimane immagazzinato, come energia, all’interno del campo elettrico del condensatore. Questa energia immagazzinata, viene restituita nella fase di scarica. Così i condensatori sono dei serbatoi di energia. • Applicazioni dei condensatori: • in una macchina fotografica, un condensatore accumula l’energia che poi, mediante una scarica veloce, fa funzionare il flash; • in un defibrillatore, un condensatore molto grande accumula l’energia che poi sarà scaricata per regolarizzare il battito cardiaco.

Esercizi Qual è l’energia immagazzinata da un condensatore, sapendo che porta una carica di 4 μ C e che tra le armature c’è una d.d.p. di 200 V ? Un condensatore di capacità 650pF è caricato fino ad ottenere una d.d.p. di 200V fra le armature. Quanto lavoro è stato compiuto per caricarlo? Tramite i due elettrodi di un defibrillatore, applicati vicino al cuore, viene scaricata una energia di 500J. La capacità del dispositivo è di 300μF. Calcola la tensione tra i due elettrodi.

ProblemA camera flash lamp stores energy in a 150 μF capacitor at 200 V. • How much electric energy can be stored? • What is the power output if nearly all this energy is released in 1.0 ms? Solution:

2. Draw: ▶ three capacitors in parallel ▶ a series of three capacitors ▶ a series made of two sets of two capacitors in parallel ▶ two parallel series of three capacitors

Conseguenze della legge di Gauss 4) La capacità di un condensatore piano è: Consideriamo come superficie gaussiana un cilindro che attraversa una delle armature come da figura. • - il flusso attraverso la superficie laterale è zero, perché la superficie è parallela al campo elettrico; • Il flusso attraverso la base di sinistra è nullo, perché il campo elettrico all’interno delle armature è zero; • l’unico contributo è quello della base di destra del cilindro, posta tra le armature; il flusso vale: cioè: Noi sappiamo che: cioè:

Conseguenze della legge di Gauss 3) Il campo elettrico tra due armature piane: Consideriamo come superficie gaussiana un cilindro che attraversa una delle armature, come da figura. • - il flusso attraverso la superficie laterale è zero, perché la superficie è parallela al campo elettrico; • Il flusso attraverso la base di sinistra è nullo, perché il campo elettrico all’interno delle armature è zero; • l’unico contributo è quello della base di destra del cilindro, posta tra le armature; il flusso vale: In base alla definizione, sappiamo che: per cui, si ha: cioè:

ELECTRIC POTENTIAL ENERGY