72. (30.2) Interaction between two parallel current carrying wires.

Transcription

1 7. (3.) Interaction between two parallel current carrying wires. Two parallel wires carrying currents exert forces on each other. Each current produces a agnetic field in which the other current is placed. B F I a I l The agnitude of the agnetic force exerted on segent l of a wire by the other wire (infinite) is F µ l II πa Parallel "currents" attract and antiparallel repel. Proof. Wire produces a agnetic field which has the following value at the other wire: r r B µ I a πa $ Therefore the agnetic force exerted on the second wire is: r F r r µ l r Il B I πa µ r r r l πa r ( I aˆ ) r µ l ( I aˆ ) I ( I I ) aˆ ) ( I I ) aˆ πa r r

2 73. (3.4, 3.5) The agnetic field of a solenoid - a long tightly wound helical coil. I N S I L Fro the syetry of the currents we can approxiate the outside field to be zero. B out and the agnetic field inside is unifor (approxiately). Its direction is parallel to the axis, and the agnitude of the fields is: Bin µ n I where n is the nuber of turns per unit length of the solenoid, and I is the current in the solenoid.

3 Proof. Iagine an aperian loop along a line of the agnetic field. The contour integral of the agnetic field along this loop is: B ds Bin ds + Bout ds B in The current through the surface enclosed by this loop is out I A N I where N is the nuber of turns in the solenoid. Fro Aper-Maxwell's equation we obtain: in L therefore Bin L µ NI N Bin µ L I µ n I 3

4 74. (3.9) Magnetic properties of atter. When a substance is placed in a (external) agnetic field ( B r ), its olecules acquire a agnetic oent related to the external field. This creates an additional agnetic field ( B r ) (internal). The agnetic state of a substance is described by the agnetization vector, M, r defined by the (internal) agnetic field of the substance. r r B µ M The net agnetic field is different when the substance is present than when it is not. Except for ferroagnetic, the agnetic susceptibility χ of the substance relates the two fields. r r r r B B + B + χ ( ) B Depending on the reaction of a aterial to an external agnetic field, the substance can be a a) ferroagnetic - a substance that produces a strong agnetic field aligned with the external field (agnetic susceptibility χ>>), which is sustained even after the external field is reoved - ferroagnetic substances exhibit hysteresis of agnetization. b) paraagnetic - a substance that produces a agnetic field aligned with the external field (χ>) c) diaagnetic - a substance that produces a agnetic field opposed to the external field (χ<). 4

5 75. (3.intro, 3., 3.) Magnetic Induction a) An induced electrootive force appears in a conducting loop when the agnetic flux through the surface enclosed by the loop is changing. This effect is called electroagnetic induction. ε Φ B N b) The electrootive force induced in a coil with N loops has value ε N d Φ dtb where Φ B is the agnetic flux through the cross-sectional surface of the coil. 76. (3.3) Lenz rule The polarity of an induced electrootive force is such that it tends to produce an induced current creating a agnetic field that opposes the change in flux causing the ef. 5

6 77. (3.4, 33.9) Mutual induction dφ Φ - + ε di I If two coils are arranged in such a way that the agnetic fluxes though cross-sectional areas of each coil are related, a changing current in one coil induces an ef in the second coil. The effect is called utual induction. The ef induced in the second coil is proportional to the rate of change in the current in the first coil. The proportionality coefficient M is called the utual inductance. ε M di dt (The SI unit of utual inductance is HVs/A) 6

7 78. (3.) Self-induction A change in current di in a coil induces an electrootive force ε L the sae coil. The ef is proportional to the rate of change in current. ε L dφ N dt N d dt ( A B) NA d dt µ A coil is an inductor with an inductance I I + ε ε L + N µ I l L µ N A l N l A di dt Voltage across an inductor is due to the induced ef only. 79. Application of electric induction (coon exaples) a) a icrophone b) the playback head of a tape deck or VCR c) an electric guitar pickup d) an AC generator e) an inductor f) a transforer g) the back ef of an electric otor h) ground fault interupter 7

8 8. (3.3) Magnetic energy a) The energy stored in an inductor depends on the inductance of the inductor and the current through the inductor: U LI Proof. The rate at which energy is delivered to the solenoid depends on the voltage across the solenoid and the current through the solenoid: P I I L di el ε dt Therefore the energy at tie t stored in the agnetic field of the solenoid is t t di I( t) U ( ) ( ) t P dt' LI t' dt' LIdI LI dt' el b) Magnetic energy density is defined as the energy stored in a agnetic field per unit volue of the field du u dv c) If, at a certain location, the agnitude of the agnetic field vector is B, the agnetic energy density at this location is: u B µ 8

9 Proof. For a solenoid Therefore u U A l LI A l B µ A l µ N L I N l A B l µ N µ B 8. (3.9) Magnetic oent of an electron. In classical physics, the agnetic oent of a particle is always associated with the angular oentu (in a circular otion) of the particle. There is evidence that an electron has a agnetic oent which is not associated with an orbital otion (Stern-Gerlach experient). The corresponding "angular oentu", called the spin, has a agnitude (very sall) of 3 S h J s The electron agnetic oent vector associated with the spin vector has the value: r e r B s Spin is an intrinsic property of an electron (like ass and charge) and should not be confused with a spinning of the electron! 9

10 8. Radiation (.7) a) The process of eitting energy in the for of waves or particles; b) Energy radiated in the for of waves or particles; 83. (34.) Electroagnetic radiation Electroagnetic radiation is associated with electroagnetic waves (oscillations of the electric and agnetic field) or beas of particles called photons. All electroagnetic waves travel in a vacuu with a speed of c 3 8 / s µ ε in all inertial fraes. In other edia, the speed of light is less and depends on the index of refraction n. The index of refraction of a ediu is defined as the ratio of the speed of light in a vacuu c to the speed of light in the ediu v. c n v Any electroagnetic radiation can be considered as a superposition of sinusoidal (haronic) electroagnetic waves.

11 84. (34.) Electroagnetic wave x k z y a) Electroagnetic waves always ove in a direction perpendicular to their electric and agnetic fields (transverse oscillations). For a onochroatic, plane wave, vector k, called the propagation vector, indicates the direction of the wave propagation. ( r, ), i cos( k r ω ) ( r, ) cos( k r ω ) E t E t i B t B t i, i In an electroagnetic wave, the electric and agnetic fields are perpendicular to each other and to the propagation vector. b) If the oscillations of the fields are haronic, the wave is onochroatic (definition). Monochroatic light produces the sensation of a defined color; but a certain color can be caused by non-onochroatic light as well. c) If the oscillations of the fields follow a certain siple pattern we say that the electroagnetic wave is polarized. When the fields oscillate along certain straight lines, the wave is linearly polarized. When the fields rotate without changing their agnitude the wave has circular polarization (an attribute of the wave).

12 85. (34.) Wave equation. a) Each of the fields satisfies the wave equation. In an epty space E x µ ε E t and B x µ ε B t proof. Applying Faraday's law E ds Φ t B to the aperian loop arked in the figure, we obtain an equation relating the agnetic and electric fields. z y B E dy dx x E ds ( + ) + ( ) + ( ) + ( ) E x dx dy E x dy E x E x dx E x dy E x dxdy Φ B t t ( Bdxdy B ) t dxdy Therefore E x B t Siilarly, fro the Apère-Maxwell law we can show B x µ ε E t

13 Cobining the last two equations leads to the wave equations E µ ε µ ε x x B B E E t t x t t t B µ ε µ ε µ ε µ ε x x E E B B t t x t t t b) The siplest solution to the wave equation is a plane wave: E( x, t) E cos ( kx ωt) B x, t B cos kx ωt ( ) ( ) proof with E B E B ω k µ ε c Recall that the wavelength and frequency of the above electroagnetic wave are λ π k and f ω π, respectively. The wave travels with a phase speed of c λf π ω k π µ ε. 3

14 Direct substitution in the wave equations k E ( x, t) µ ε µ ε ω E( x, t) x E t E leads to ω k µ ε Through a direct substitution is a differential equation relating agnetic and electric field kb sin B x E t ( kx ωt) µ ε µ ε ωe sin ( kx ωt) we obtain E B E k B µ ε ω µ ε 4

15 86. (34.3) The energy carried by electroagnetic waves a) The (total) energy density u of an electroagnetic wave is defined as the energy du of both the electric and agnetic fields per unit volue of the wave. (SI unit: J/ 3 ) u du dv The average energy density coprised by a sinusoidal wave depends on the aplitudes of the agnetic and electric fields uav εe εers uav B Brs µ µ where E rs and B rs are the root-ean-square values of the electric and agnetic fields respectively. proof. The (instantaneous) total energy density at a certain location is equal to the su of the energy density associated with each field separately u( t) ε E B εe E + + ε µ µ c E 5

16 Fro the definition of average value of a function we find u av t t u t dt li li E t t t t li ε t t t' + sin ( kx ωt' ) ε E E ω li ε t t cos t' dt' t ( ') ' εe cos ( kx ωt) dt' + ( kx ω ) t b) The rate of energy flow across a unit area in electroagnetic wave is described by the Poynting vector defined by the following expression S E B µ proof. Consider, a plane electroagnetic wave. If we iagine a surface (with area da) perpendicular to the path of the wave, the aount of energy passing this surface in a differential tie dt is ( da cdt) du u where u(x,t) is the instantaneous energy density at the surface. The rate at which the energy is transferred across the surface is therefore du dt (, ) c u x t da 6

17 Considering the agnitude of the Poynting vector at the location of the surface we find (, ) S c u( x, t) S x t EB E µ ε E ε E µ cµ µ µ ε da Note that the direction of the Poynting vector coincides with the direction of the wave propagation while its agnitude is equal to the rate of energy transfer per unit area. c) Wave intensity I is the ratio of the average rate at which an electroagnetic wave carries energy through an iaginary surface, which is perpendicular to the propagation direction, to the area of the surface. du dt (SI unit W/ ) I S av d) Intensity I and energy density u of a wave are proportional with the speed of light being the proportionality constant. ( ( )) I S c u t cu av av av For a certain electroagnetic radiation (a superposition of sinusoidal waves), the function relating the intensity of each sinusoidal wave with the wavelength is called the spectru of that radiation. 7

18 87. (34.7) Electroagnetic radiation that stiulates the huan eye is called visible light. This includes waves with wavelengths between 4n and 7n. The axiu sensitivity of the huan eye is to light with a wavelength of about 555n, which corresponds to the axiu intensity of the sun's spectru. Electroagnetic radiation also includes radio waves, infrared and ultraviolet (black) light, X-rays and γ-rays. 8

19 LIGHT AND OPTICS 88. (35.3, 35.4, 35.5) Reflection and refraction At the boundary of two edia, light undergoes both reflection and refraction. incident ray angle of incidence θ θ angle of reflection reflected ray θ angle of refraction refracted ray a) The law of reflection The reflected ray lies in the plane of incidence and the angle of reflection θ ' is equal to the angle of incidence θ θ θ ' b) The law of refraction (Snell's law) The refracted ray lies in the plane of incidence and the angle of refraction θ is related to the angle of incidence by n sin θ n sin θ In general, index of refraction of a ediu is a function of the radiation wavelength. 9

20 89. (35.7) Total internal reflection Total internal reflection occurs when light travels fro a ediu of high index of refraction to one of lower index of refraction. The sallest angle θc at which total internal reflection occurs is called the critical angle. n θ c sin n ( n n ) > because nsin θ c nsin9 9. In order to be seen, an object ust send light fro each of its points in any directions. The eye collects soe of the light eitted fro a point allowing the brain to interpret the location of the point. In soe situations diverging rays are interpreted as originating fro a single point creating an iage of a point.