review Problem 23.83

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1 review Problem A metal sphere with radius R 1 has a charge Q 1. (a) What are the electric field and electric potential at the surface of the sphere? Take the potential to be zero at an infinite distance away. The sphere is now connected by a long, thin, conducting wire to another sphere of radius R 2 that is several metres from the first sphere. Before the connection is made, this second sphere is uncharged. After electrostatic equilibrium has been reached, what are (b) the total charge on each sphere? (c) the electric potential at the surface of each sphere? (d) the electric field at the surface of each sphere? Assume that all of the charge is on the spheres, and none is on the wire. PHYS153_08W 1

2 Problem In the circuit shown in the figure, each capacitor initially has a charge of magnitude 3.5 nc on its plates. After the switch S is closed, what will be the current in the circuit at the instant that the capacitors have lost 80% of their initial energy? Fig PHYS153_08W 2

3 Magnetic field and magnetic forces Modern understanding of magnetostatics began only in : Oersted discovered that an electric current deflected a compass needle (and hence produced a magnetic field). 1831: Faraday and Henry (independently) found that a changing magnetic field produces an electric field. 1860: J.C. Maxwell developed a complete theory of electricity and magnetism that showed that a changing electric field produces a magnetic field. Fig : Heinrich Hertz demonstrated the existence of E.M. waves. 1905: Einstein: Special Theory of Relativity. PHYS153_08W 3

4 We now know that magnetic phenomena result from relativistic effects of charges in motion. In the words of E.M. Purcell (Nobel Prize winning physicist) The magnetic interaction of electron currents can be recognized as an inevitable corollary of Coulomb s law. If the postulates of relativity are valid, if electric charge is invariant, and if Coulomb s law holds, then the effects we call magnetic are bound to occur. Historically, formulae were derived to describe magnetic phenomena. It is easier to use these then to go through the complexities of relativistic corrections each time. mks unit Tesla (1 T) cgs unit Gauss (G) 1 Tesla = 10 4 Gauss. 0.5G Earth s magnetic field Field of sunspots few hundred G. Iron electromagnet 20kG Superconductivity magnet kG Interstellar magnetic field of the Galaxy 10µ G PHYS153_08W 4

5 Note Fig on slide 3. Earth acts as if it had a giant magnet inside. Earth s magnetic dynamo not competely understood. Can you think of an advantage of living on a planet having a magnetic field? Magnetic forces on moving charges. --found by experiment that F B q, B, v. --found by experiment that F B not in the direction of B, but is perpendicular to both B and the velocity vector v. F =qvbsinφ or F = qvxb (use R.H. rule for a positive charge.) Note the directions of v, B, and F in the figure. The R.H. rule puts the thumb in the direction of F. (Another unit of B is N/A.m) Fig PHYS153_08W 5

6 Note the differences between Electricity and Magnetism. there are single electric charges but no monopoles. Electric forces are generated by static charges (as well as moving). Magnetic forces are generated only by moving charges. F e is in the direction of E, but F B is not in the direction of B. If a charged particle moves through a region of space where there is both an electric field and a magnetic field, F = q( E+ vxb) (Lorentz force) Question: A uniform B field is directed into the plane of the slide, as shown. A negatively charged particle moves from left to right in the plane. which path does it follow, 1, 2, or 3? (1) x x x x x x x x x x x x x x x x x x x x x x x x x x x (3) (2) PHYS153_08W 6

Note the configuration of field lines for several common sources of magnetic field.

7 Unlike electric field lines, magnetic field lines are not lines of force. The force on a charge moving in a B field is perpendicular to both B and v. Note the configuration of field lines for several common sources of magnetic field. bar magnet on slide 3 current carrying wire in Fig b. Direction of B given by a R.H. rule for positive current. solenoid in Fig c. Fig b Fig c PHYS153_08W 7

Magnetic Flux Recall the ideas associated with electric flux. Magnetic flux involves some of the same ideas. In Fig. 27.15, what is the magnetic flux through the surface?

8 Magnetic Flux Recall the ideas associated with electric flux. Magnetic flux involves some of the same ideas. In Fig , what is the magnetic flux through the surface? For area da, dφ B = B da = BcosφdA= B da Fig and Φ = B BcosφdA = B da If B is uniform over a plane surface as in Fig a, Then the flux is given by Φ B =BAcosφ In this figure, φ = o o = As in the case of electric flux, is a maximum when, Φ B =0 φ and when. 60 o (Wb) φ ΦB = 0 o =90 Fig a PHYS153_08W 8 φ A

There is one big difference between electric flux and magnetic flux. For electric flux through a closed surface, E da=0 when the net charge enclosed is zero.

9 There is one big difference between electric flux and magnetic flux. For electric flux through a closed surface, E da=0 when the net charge enclosed is zero. For magnetic flux, there is no magnetic monopole. Hence there can t be any net magnetic charge or magnetic pole enclosed. There are only dipoles. So, in the case of magnetic flux, B da=0 (magnetic flux through any (always true) closed surface) (More on units: 1Wb=1T.m 2 ; and since 1T=1N/A.m, 1Wb=1N.m/A) Charged particles moving in a magnetic field For a charged particle moving in a magnetic field, the force on it is given by = F qvxb The direction of the force on a positive charge Is given by the R.H. rule. F is perpendicular to v and B as shown in the figure. Fig PHYS153_08W 9

10 Hence the magnetic force never does work on the particle. The velocity will remain constant in magnitude. The particle will move in a circle. F = qvb 2 v = m R And the radius R is given by R= mv qb The angular speed of the particle is given by ω= v = R qb m For a given particle in a given magnetic field, this is a constant, and is called the cyclotron frequency. x x x x x x x x x B in x x x x x x x x x x In the cyclotron accelerator (to which x x x x x x x x x x E class Triumf belongs), positive charges q (q) are accelerated by an electric field across the gaps between the D s and gain K.E. E x x x x x x x x x They then enter the region of a magnetic x x x x x x x x x field inside the hollow chamber (one side x x x x x x x of the D s and move in a circle. PHYS153_08W 10

The radio frequency field producing the electric field in the gap has a frequency equal to ½ the cyclotron frequency so that the particles are accelerated every time they cross a gap.

11 The radio frequency field producing the electric field in the gap has a frequency equal to ½ the cyclotron frequency so that the particles are accelerated every time they cross a gap. After reaching a certain energy, the particles are deflected out of orbit to hit a target. When a charged particle enters a magnetic field with velocity components v and v, the parallel component Is unaffected by B. The particle will now move in a helix. The parallel component of velocity will be constant in the direction of B (Fig ). Fig The earth s non-uniform field traps charged particles from the sun. They are trapped in the Van Allen radiation belts (Fig ), discovered by Van Allen in When these particles leak out of the fields at the poles, they are the source of the aurora borealis. Fig PHYS153_08W 11

Fig. 27.19 shows a non-uniform field generated by two coils separated by some distance. The field is uniform right near the coils, but bulges in the middle.

Particles to the right of the bulge, experience a force to the left. Particles near either ring experience a force toward the centre.

12 Fig shows a non-uniform field generated by two coils separated by some distance. The field is uniform right near the coils, but bulges in the middle. This can be used to illustrate how particles behave in the earth s field. Particles to the left of the bulge experience a force to the right. Particles to the right of the bulge, experience a force to the left. Particles near either ring experience a force toward the centre. Some particles will spiral back and forth many times. In the case of the earth s field, some particles will eventually leak out near the poles. Fig Thomson s e/m experiment J.J. Thomson 1897 in the Cavendish Laboratory. This is one of the hallmark experiments in physics. PHYS153_08W 12

13 Nothing was known about electrons at the time. Electrons are accelerated by a potential difference V between the anodes mv =ev v = 2eV m The next region between plates P and P has crossed E and B fields which act as a velocity filter. Magnetic force on the electron, evb, is down. Electrostatic force, ee, is up. v= E B So, when the particles pass through undeviated. By equating these two expressions for v, we obtain e = m 2 E 2VB 2 (=1.759 x C/kg) By varying E, V, B, Thomson found only one value for e/m. PHYS153_08W 13

14 This showed that electrons were a common constituent of all matter. In 1912, Millikan measured the value of the charge e in his famous oil drop experiment. This then allowed the determination of m. e=1.602 x C m=9.109 x kg Mass Spectrometer --used to measure atomic and molecular masses. Ions (from a source at top of diagram) pass through slits S 1 and S 2 to form a narrow beam. The ions pass through a velocity filter with crossed E and B fields. The ions that emerge will have v = E/B. The ions pass into a region where there is a magnetic field, B. They will move in circular arcs of radius R = mv/qb. Most ions will have a charge of +e. The mass m can then be measured. Fig Bainbridge mass spectrometer PHYS153_08W 14

15 Problem The magnetic flux through one face of a cube is Wb. (a) What must the total magnetic flux through the other five faces of the cube be? One possible scenario Fig adapted PHYS153_08W 15

Magnetic Fields. or I in the filed. ! F = q! E. ! F = q! v! B. q! v. Charge q as source. Current I as source. Gauss s Law. Ampere s Law.

Magnetic Fields. or I in the filed. ! F = q! E. ! F = q! v! B. q! v. Charge q as source. Current I as source. Gauss s Law. Ampere s Law. Magnetic Fields Charge q as source Gauss s Law Electric field E F = q E Faraday s Law Ampere-Maxwell Law Current I as source Magnetic field B Ampere s Law F = q v B Force on q in the field Force on q v