Gravitational & Electric Fields

Transcription

1 Gravitational & Electric Fields Jessica Wade Department of Physics & Centre for Plastic Electronics, Imperial College London Faculty of Natural & Mathematical Sciences, King s College London

2 Gravitational Field Newton: there is an attractive force between all objects with mass Forces always occur in pairs: object is pulled by earth, earth is pulled by object Uniform gravitational field: Gravitational force (F G ) = mass acceleration of free fall F 0 = m g Gravitational field strength = 3 4 = 56 = g 5 5 What is weight? Weight of an object is the gravitational force exerted on that object by the mass of the Earth

3 Non-uniform gravitational fields Newton proposed that the strength of the Earth s gravitational field varies inversely with the square from its centre F = G m :m ; r ; m : g = G m :M GM r ; g = r ; G = Gravitational Constant = 6.67 x Nm 2 kg -1

4 Gravitational Fields Work is done when a force moves something Work done = Force x Distance moved in the direction of the force A system has energy if it is capable of doing work Gravitational Potential Energy = W = mgh The total energy of a system is conserved Change in GPE = E = mg h

5 Gravitation Potential in a Radial Field Why is gravitational potential energy negative? Object of mass m in empty space, r away from any other massive body Force = GMm/r 2, but as it is at infinity, there are 0 forces acting upon it Cannot fall toward anything à no potential energy à cannot do any work (GPE = 0) Mass m now sits on Earth To move away, give it energy. Gets to infinity = 0 GPE Only way to balance is to say it has negative GPE on Earth

6 GPE in Radial Fields F = G Mm D E E A = GMm B 1 r D ; dr = GMm 1 1 r F ; r : Gravitational potential is the change in potential energy for a unit mass that moves from infinity to a point at less than infinity (m = 1) V = GM r r ;

7 Variations of G with r V = GM r

8 Variations of g with r g = V r

9 Energy of an asteroid falling to Earth Kinetic energy of an asteroid falling to earth Loss of gravitational potential = 0N D O per unit mass Gain in KE = loss in GPE GPE = GP x mass of asteroid E P = : ; mv; = 0N O5 D O Can also calculate escape velocty from massive body: v RST = 2 GM V r V

10 Satellites in Orbit A satellite moves in a circular orbit with an inward gravitational acceleration g and speed v: g = v; r Speed of a satellite, v = gr Geostationary/ Geosynchronous satellites stay still relative to Earth Circumference = 2πr S = vt Where r S = orbital radius, t = 24 hours = 24 x 60 x 60 seconds

11 Geostationary Satellites v = 2πr S t v = 2πr S t v = gr S g = GM r ; S = GM r S ; r S = GM r S

12 Electric Fields Charged bodies exert a force on each other: Any charged body in the space around another charged body is acted on by an electric field The field between two parallel charged plates is uniform What is the definition of Electric Field Strength? Electric field strength = Force [N] on each coulomb of charge

13 Moving Charges Work done by a force of F moving through plates of separation d Work Done = Q V = F d [F]=N, [d]=m, [Q]=C, [V]=V=JC -1 F Q = V d The magnitude of a uniformelectric field: E = V d [E]=V m -1 Calculating the speed of moving charges from an electron gun Thermionic Emission: Electrons with enough energy escape the surface of the wire Charges accelerate between filament and anode, gaining KE in E: KE = : ; mv ; = ev

14 Direction of an Electric Field Direction of a positive charge (from positive to negative) Strength of field = spacing of lines Arrows on lines = direction of electric field Parallel, evenly spaced lines = uniform electric field strength

15 Electric Potential The voltage measured in the field between two plates is the electric potential Electric potential is the potential difference between the 0 V plate and the probe (voltmeter) Equipotentials are always at right angles to field lines Take care at corners of plates where field no longer uniform Field strength = potential gradient E = dv dr

16 Parallel Plate Capacitor Electric field between two parallel plates can store charge (capacitor) Charge on plates potential difference CV Charge on plates area plates Q A V d Medium between plates (dielectric) is an insulator Q A = ε V b d ε b is the permittivity of free space, [ε b ] = F m -1 A 1 farad capacitor charged by a potential difference of 1 volt carries a charge of 1 coloumb

17 Parallel Plate Capacitor Capacitance: C = Q V = ε A b d A C = ε b ε D d ε b is the relative permittivity of the medium ε d air = 1, paper = 2 3, water = 80

18 Non-Uniform Electric Fields Coulomb s Law: Force depends on Q 1, Q 2 and r: F = k Q :Q ; r ; Notice any similarities? F = G m :m ; r ; Gravity = always attractive Electric = attractive/ negative Electric Field Strength considers force on a test charge Q 2 at a distance r from Q 1 E = kq :Q ; 1 r ; = kq : Q ; r ;

19 Potential in a radial field V = kq : r Find k: isolated charged sphere, where radius = r and charge = Q Where A = 4πr ; Q A = ε b V d = ε bkq r ; Q 4πr ; = ε bkq r ; k = 1 4πε b

20 Potential in a radial field Electric field = potential gradient = fg V = 1 Q 4πε b r E = 1 d : D 4πε b dr E = Q 4πε b r ; fd

21 Comparing Electric and Gravitational Fields Gravitational Electric Force F = G m 1m 2 r 2 F = k Q 1 Q 2 r 2 Field Strength E = G m r 2 E = Q 4πε 0 r 2 Potential V = G m r V = 1 Q 4πε 0 r