Physics / Higher Physics 1A. Electricity and Magnetism Revision

Transcription

1 Physics / Higher Physics 1A Electricity and Magnetism Revision

2 Electric Charges Two kinds of electric charges Called positive and negative Like charges repel Unlike charges attract

3 Coulomb s Law In vector form, F 12 k e q 1 q 2 r ˆ is a unit vector r 2 ˆ r directed from q 1 to q 2 Like charges produce a repulsive force between them

4 The Superposition Principle The resultant force on q 1 is the vector sum of all the forces exerted on it by other charges: F 1 = F 21 + F 31 + F 41 +

5 Electric Field Continuous charge distribution 2 ˆ e e o q k q r F E r ˆ ˆ lim i i e i e q i i q d q k k r r E r r

6 Electric Field Lines Dipole The charges are equal and opposite The number of field lines leaving the positive charge equals the number of lines terminating on the negative charge

7 Electric Flux E E i A i cos i E i A i lim E A d E i i A 0 i E A s u r f a c e

8 Gauss s Law E E d A q in q in is the net charge inside the surface E represents the electric field at any point on the surface 0

9 Field Due to a Plane of Charge The total charge in the surface is σa Applying Gauss s law E 2 EA A, and E Field uniform everywhere

10 Properties of a Conductor in Electrostatic Equilibrium 1. Electric field is zero everywhere inside conductor 2. Charge resides on its surface of isolated conductor 3. Electric field just outside a charged conductor is perpendicular to the surface with magnitude σ/ε o 4. On an irregularly shaped conductor surface charge density is greatest where radius of curvature is smallest

11 Electric Potential Energy Work done by electric field is F. ds = q o E. ds Potential energy of the charge-field system is changed by ΔU = -q o E. ds For a finite displacement of the charge from A to B, the change in potential energy is U U U q E d s B A o B A

12 Electric Potential, V The potential energy per unit charge, U/q o, is the electric potential U B V E A q o d s The work performed on the charge is W = ΔU = q ΔV In a uniform field B A B V V V E d s E d s E d A B A

13 Equipotential Surface Any surface consisting of a continuous distribution of points having the same electric potential For a point charge V k e q r

14 Finding E From V From V = -E.ds = -E x dx E x dv dx Along an equipotential surfaces V = 0 Hence E ds i.e. an equipotential surface is perpendicular to the electric field lines passing through it

15 V Due to a Charged Conductor E ds = 0 So, potential difference between A and B is zero Electric field is zero inside the conductor So, electric potential constant everywhere inside conductor and equal to value at the surface

16 Cavity in a Conductor Assume an irregularly shaped cavity is inside a conductor Assume no charges are inside the cavity The electric field inside the conductor must be zero

17 Definition of Capacitance The capacitance, C, is ratio of the charge on either conductor to the potential difference between the conductors C Q V A measure of the ability to store charge The SI unit of capacitance is the farad (F)

18 Capacitance Parallel Plates Charge density σ = Q/A Electric field E = / 0 (for conductor) Uniform between plates, zero elsewhere C Q V Q Ed Q Q 0 A d 0 A d

19 Capacitors in Parallel Capacitors can be replaced with one capacitor with a capacitance of C eq C eq = C 1 + C 2

20 Capacitors in Series Potential differences add up to the battery voltage Q Q 1 Q 2 V V 1 V 2 V Q V 1 Q 1 V 2 Q 2 1 C 1 C 1 1 C 2

21 Energy of Capacitor Work done in charging the capacitor appears as electric potential energy U: 2 U Q 1 Q V 1 C ( V ) 2 C 2 2 Energy is stored in the electric field Energy density (energy per unit volume) u E = U/Vol. = ½ o E 2 2

22 Capacitors with Dielectrics A dielectric is a nonconducting material that, when placed between the plates of a capacitor, increases the capacitance For a parallel-plate capacitor C = κc o = κε o (A/d)

23 Rewiring charged capacitors Two capacitors, C 1 & C 2 charged to same potential difference, V i. Capacitors removed from battery and plates connected with opposite polarity. Switches S 1 & S 2 then closed. What is final potential difference, V f?

24 Q 1i, Q 2i before; Q 1f, Q 2f after. Q 1i = C 1 V i ; Q 2i = -C 2 V i So Q=Q 1i +Q 2i =(C 1 -C 2 )V i But Q= Q 1f +Q 2f (charge conserved) With Q 1f = C 1 V f ; Q 2f = C 2 V f hence Q 1f = C 1 /C 2 Q 2f So, Q=(C 1 /C 2 +1) Q 2f With some algebra, find Q 1f = QC 1 /(C 1 +C 2 ) & Q 2f = QC 2 /(C 1 +C 2 ) So V 1f = Q 1f / C 1 = Q / (C 1 +C 2 ) & V 2f = Q 2f / C 2 = Q / (C 1 +C 2 ) i.e. V 1f = V 2f = V f, as expected So V f = (C 1 - C 2 ) / (C 1 + C 2 ) V i, on substituting for Q

25 Magnetic Poles Every magnet has two poles Called north and south poles Poles exert forces on one another Like poles repel N-N or S-S Unlike poles attract N-S

26 Magnetic Field Lines for a Bar Magnet Compass can be used to trace the field lines The lines outside the magnet point from the North pole to the South pole

27 Direction F B perpendicular to plane formed by v & B Oppositely directed forces are exerted on charges of different signs cause the particles to move in opposite directions

28 Direction given by Right-Hand Rule Fingers point in the direction of v (for positive charge; opposite direction if negative) Curl fingers in the direction of B Then thumb points in the direction of v x B; i.e. the direction of F B

29 The Magnitude of F The magnitude of the magnetic force on a charged particle is F B = q vb sin is the angle between v and B F B is zero when v and B are parallel F B is a maximum when perpendicular

30 Force on a Wire F = I L x B L is a vector that points in the direction of the current (i.e. of v D ) Magnitude is the length L of the segment I is the current = nqav D B is the magnetic field

31 Force on a Wire of Arbitrary Shape The force exerted segment ds is F = I ds x B The total force is a b F I d s B

32 Force on Charged Particle Equating the magnetic & centripetal forces: F qvb mv 2 Solving gives r = mv/qb r

33 Biot-Savart Law db is the field created by the current in the length segment ds Sum up contributions from all current elements I.ds B 0 4 I d s ˆ r r 2

34 B for a Long, Straight Conductor B 0 I 2 a

35 B for a Long, Straight Conductor, Direction Magnetic field lines are circles concentric with the wire Field lines lie in planes perpendicular to to wire Magnitude of B is constant on any circle of radius a The right-hand rule for determining the direction of B is shown Grasp wire with thumb in direction of current. Fingers wrap in direction of B.

36 Magnetic Force Between Two Parallel Conductors F 1 0 I 1 I 2 2 a l Parallel conductors carrying currents in the same direction attract each other Parallel conductors carrying currents in opposite directions repel each other

37 Definition of the Ampere The force between two parallel wires can be used to define the ampere F 1 l 0 I 1 I 2 2 a with T m A -1 When the magnitude of the force per unit length between two long parallel wires that carry identical currents and are separated by 1 m is 2 x 10-7 N/m, the current in each wire is defined to be 1 A

38 Ampere s Law The line integral of B. ds around any closed path equals o I, where I is the total steady current passing through any surface bounded by the closed path. B ds 0 I

39 Field in interior of a Solenoid Apply Ampere s law The side of length l inside the solenoid contributes to the field Path 1 in the diagram B d s B d s B path 1 path 1 ds B B 0 N l I 0 ni

40 Ampere s vs. Gauss s Law B ds 0 I E da q 0 Integrals around closed path vs. closed surface. i.e. 2D vs. 3D geometrical figures Integrals related to fundamental constant x source of the field. Concept of Flux the flow of field lines through a surface.

41 Gauss Law in Magnetism Magnetic fields do not begin or end at any point i.e. they form closed loops, with the number of lines entering a surface equaling the number of lines leaving that surface Gauss law in magnetism says: B B.d A 0

42 Faraday s Law of Induction The emf induced in a circuit is directly proportional to the rate of change of the magnetic flux through that circuit N d B dt QuickTime and a Cinepak decompressor are needed to see this picture.

43 Ways of Inducing an emf d dt BA cos Magnitude of B can change with time Area enclosed, A, can change with time Angle can change with time Any combination of the above can occur

44 Motional emf Motional emf induced in a conductor moving through a constant magnetic field Electrons in conductor experience a force, F B = qv x B that is directed along l In equilibrium, qe = qvb or E = vb

45 Sliding Conducting Bar Magnetic flux is The induced emf is d B dt Thus the current is d dt B Blx Blx Bl dx dt Blv I R Blv R

46 Induced emf & Electric Fields A changing magnetic flux induces an emf and a current in a conducting loop An electric field is created in a conductor by a changing magnetic flux Faraday s law can be written in a general form: E.ds d B dt Not an electrostatic field because the line integral of E. ds is not zero.

47 Generators Electric generators take in energy by work and transfer it out by electrical transmission The AC generator consists of a loop of wire rotated by some external means in a magnetic field

48 Rotating Loop Assume a loop with N turns, all of the same area, rotating in a magnetic field The flux through one loop at any time t is: B = BA cos = BA cos wt N d B dt NAB d dt cos w t NAB w sin w t

49 Motors Motors are devices into which energy is transferred by electrical transmission while energy is transferred out by work A motor is a generator operating in reverse A current is supplied to the coil by a battery and the torque acting on the current-carrying coil causes it to rotate

50 Eddy Currents Circulating currents called eddy currents are induced in bulk pieces of metal moving through a magnetic field From Lenz s law, their direction is to oppose the change that causes them. The eddy currents are in opposite directions as the plate enters or leaves the field

51 Equations for Self-Inductance Induced emf proportional to the rate of change of the current L L di dt L is a constant of proportionality called the inductance of the coil.

52 Inductance of a Solenoid Uniformly wound solenoid having N turns and length l. Then we have: N B 0 ni 0 I l NA B BA 0 l I L N B I 0 N 2 A l

53 Energy in a Magnetic Field Rate at which the energy is stored is d U d I L I d t d t I U L IdI 1 2 LI 2 0 Magnetic energy density, u B, is u B U Al B 2 2 0

54 RL Circuit I R 1 e Rt L R 1 e t t Time constant, tl / R, for the circuit t is the time required for current to reach 63.2% of its max value