Module 9: Electomagnetic Waves-I Lectue 9: Electomagnetic Waves-I What is light, paticle o wave? Much of ou daily expeience with light, paticulaly the fact that light ays move in staight lines tells us that we can think of light as a steam of paticles. This is futhe bone out when place an opaque object in the path of the light ays. The shadow, as shown in Figue 9.1 is a pojected image of the object, which is what we expect if light wee a steam of paticles. But a close look at the edges of the shadow eveals a vey fine patten of dak and bight bands o finges. Such a patten can also be seen if we stetch out ou hand and look at the sky though a thin gap poduced by binging two of ou finges close. This cannot be explained unless we accept that light is some kind of a wave. It is now well known that light is an electomagnetic wave. We shall next discuss what we mean by an electomagnetic wave o adiation. 9.1 Electomagnetic Radiation 00 00 11 11 0 0 1 1 00 0 11 1 Figue 9.1: Shadow of an opaque object. 55
56 CHAPTER 9. ELECTROMAGNETIC WAVES-I P q Figue 9.2: A chage placed at a distance fom the point of obsevation P. What is the electic field poduced at a point P by a chage q located at a distance as shown in Figue 9.2? Anybody with a little knowledge of physics will tell us that this is given by Coulomb s law E = q 4πǫ 0 ê 2 (9.1) whee ê is an unit vecto fom P to the position of the chage. In what follows we shall follow the notation used in Feynman Lectues ( Vol. I, Chapte 28, Electomagnetic adiation). In the 1880s J.C. Maxwell poposed a modification in the laws of electicity and magnetism which wee known at that time. The change poposed by Maxwell unified ou ideas of electicity and magnetism and showed both of them to be manifestations of a single undelying quantity. Futhe it implied that Coulomb s law did not tells us the complete pictue. The coect fomula fo the electic field is E = q 4πǫ 0 [ê d + 2 c dt ) (ê 2 + 1 d 2 ] c 2 dt 2ê (9.2) This fomula incopoates seveal new effects. The fist is the fact that no infomation can popagate instantaneously. This is a dawback of Coulomb s law whee the electic field at a distant point P changes the moment the position of the chage is changed. This should actually happen afte some time. The new fomula incopoates the fact that the influence of the chage popagates at a speed c. The electic field at the time t is detemined by the position of the chage at an ealie time. This is efeed to as the etaded position of the chage, and ê also efes to the etaded position. The fist tem in eq. (9.2) is Coulomb s law with the etaded position. In addition thee ae two new tems which aise due to the modification poposed by Maxwell. These two tems contibute only when the chage moves. The magnetic field poduced by the chage is B = ê E/c (9.3)
9.1. ELECTROMAGNETIC RADIATION 57 E a Figue 9.3: Electic field fo an acceleated chage. A close look at eq. (9.2) shows that the contibution fom the fist two tems falls off as 1/ 2 and these two tems ae of not of inteest at lage distances fom the chage. It is only the thid tem which has a 1/ behaviou that makes a significant contibution at lage distances. This tem pemits a chaged paticle to influence anothe chaged paticle at a geat distance though the 1/ electic field. This is efeed to as electomagnetic adiation and light is a familia example of this phenomenon. It is obvious fom the fomula that only acceleating chages poduce adiation. The intepetation of the fomula is substantially simplified if we assume that the motion of the chage is elatively slow, and is esticted to a egion which is small in compaison to the distance to the point whee we wish to calculate the electic field. We then have d 2 d2 = dt2ê dt 2 ( ) (9.4) whee is the acceleation of the chage in the diection pependicula to ê. The paallel component of the acceleation does not effect the unit vecto ê and hence it does not make a contibution hee. Futhe, the motion of the chage makes a vey small contibution to in the denominato, neglecting this we eplace with the constant distance. The electic field at a time t is elated to a(t /c) which is the etaded acceleation as E(t) = q a(t /c) sin (9.5) 4πǫ 0 c 2 whee is the angle between the line of sight ê to the chage and the diection of the etaded acceleation vecto. The electic field vecto is in the diection obtained by pojecting the etaded acceleation vecto on the plane pependicula to ê as shown in Figue 9.3. Poblem 1: Show that the second tem inside the backet of eq.(9.2) indeed falls off as 1/ 2. Also show that the expession fo electic field fo an acceleated chage i.e. eq. (9.5) follows fom it.
58 CHAPTER 9. ELECTROMAGNETIC WAVES-I P e^ dz q e^ d^e dz/ d^e 90 d e^ = dzsin / Figue 9.4: Solution to Poblem 1. Solution 1: See fig. 9.4 ( and can be teated as constants with espect to time). 9.2 Electic dipole adiation. We next conside a situation whee a chage acceleates up and down along a staight line. The analysis of this situation using eq. (9.5) has wide applications including many in technology. We conside the device shown in Figue 9.5 which has two wies A and B connected to an oscillating voltage geneato. Conside the situation when the teminal of the voltage geneato connected l A V B Figue 9.5: An electic dipole.
9.2. ELECTRIC DIPOLE RADIATION. 59 D G Figue 9.6: A dipole with a detection. to A is positive and the one connected to B is negative. Thee will be an accumulation of positive chage at the tip of the wie A and negative chage at the tip of B espectively. The electons ush fom B to A when the voltage is evesed. The oscillating voltage causes chage to oscillate up and down the two wie A and B as if they wee a single wie. In the situation whee the time taken by the electons to move up and down the wies is much lage than the time taken fo light signal to coss the wie, this can be thought of as an oscillating electic dipole. Note that hee we have many electons oscillating up and down the wie. Since all the electons have the same acceleation, the electic fields that they poduce adds up. The electic field poduced is invesely popotional to the distance fom the oscillato. At any time t, the electic field is popotional to the acceleation of the chages at a time t /c in the past. It is possible to measue the adiation using a anothe electic dipole oscillato whee the voltage geneato is eplaced by a detecto, say an oscilloscope. An applied oscillating electic field will give ise to an oscillating cuent in the wies which can be conveted to a voltage and measued. A dipole can measue oscillating electic fields only if the field is paallel to the dipole and not if they ae pependicula. A dipole is quite commonly used as an antenna to eceive adio waves which is a fom of electomagnetic adiation. Figue 9.6 shows an expeiment whee we use a dipole with a detecto (D) to measue the electomagnetic adiation poduced by anothe oscillating electic dipole (G). The detected voltage is maximum at = 90 and falls as sin in othe diections. At any point on the cicle, the diection of the electic field vecto of the emitted adiation is along the tangent. Often we find that placing the antenna of a tansisto adio in a paticula oientation impoves the eception. This is oughly aligning the antenna with the incoming adiation which was tansmitted by a tansmitte.