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2. Four identical dipole radiators are aligned parallel to one another and are equally spaced along a line at a distance apart. They are driven at a frequency of and are phased so that, starting from one end, each successive dipole lags the preceding one by . Find the intensity pattern of the radiation at a great distance in the equatorial plane (perpendicular to the dipole axes), and sketch this function in polar coordinates. Such a diagram is called the radiation pattern of an antenna system. My ans. At fixed point in equatorial plane with angle from dipole radiators aligned line, Let phase diff each adjacent dipole = : , where . first method Field, Using complex, Intensity, , substitue (1) Notice Above eq.(1) is valid when i.e. When , from orignal Intensity eq. second method just using eq.(30.3) substitue , graph x: red, y: green Initial intensity :1 Intensity scale : 1000 distance > 100 This could...

1. Two antennas are arranged as shown in Fig. 29-5(a) and are driven in phase. The antennas are driven so that one would, if alone, radiate a certain intensity in all horizontal directions, and the other, an intensity . What should the observed intensities in the various directions shown in the figure be? My ans. Intensity is proportional to square of the field. So, means . N, S phase diff = , N60, S60 using eq.(29.17) phase diff = using eq. (29.16) substitute, E, W in phase, This could be wrong.

1. Interpret the following two problems in complex numbers geometrically, and thus find absolute value of A in each case a) b) Ans a) b) Let radius of circle R(r in fig., but we have already r in problem. ), —-(1) —-(2) substitute (1) to (2)

Online books or notes Mathematical Tools for Physics The Poor Man’s Introduction to Tensors

Newtonian Dynamics by By Richard Fitzpatrick At UT Austin undergraduate senior level

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For fast convergence, it can be written without first term(i.e. 1) Using , bounds from integral test becomes, For six significant figures, at least , which gives where is negligible. So, six significant figures can be obtained from . , where bounds are check with simple javascript k= constant= ,bound{ }

For what values of p and q will converge? For sufficiently large independent of and , , so is dominant. , converges, so converges for all . , diverges(it’s harmonic numbers). But using integral test, , range of can be divided. converges. diverges. diverges. , diverges, so diverges for all .