PHY323 - Electromagnetic Theory at Farmingdale State

Assignment Task

Questions

1. Recall the expression of the electric and magnetic field due to a charge particle moving in an arbitrary trajectory from the exercise given in the lecture note.

(a) Show that the Poynting vector is S = c 4π [E 2Rˆ − (Rˆ E)E]

Where Rˆ is the unit vector along the vector connecting the observation point and the retarded position of the charge.

(b) Now suppose at retarded time tr, the instantaneous velocity of the charge is zero. Show that the Poynting vector is S = cq2a 2 4πR2 sin2 θ

Where a is the acceleration of the charge and θ is the angle between Rˆ and a

(c) Find the power radiated in the direction Rˆ i.e., dP dΩ. Find the total power radiated.

(d) Now suppose the particle moves as z(t) = a cos(ωt)

Find time averaged dP dΩ for this case and plot θ vs dP dΩ.

2. Recall that the field strength that we obtain from the Lienard Wiechart potential is:

• Fµν = − q r 2k[µaν] + 2k · ak[µuν] + 2q r 2 k[µuν]

3. Consider a charged particle of charge q moving along a circle of radius a with a uniform speed v. Assume v << c>

4. Consider two charged particles of the same charge q moving along a circle of radius a with a uniform speed v such that they are always diametrically opposite to each other. Obtain the electric field E~ and and magnetic field B~ at a point in the wave zone. Find the time average of the angular distribution and the total power radiated by the system.

5. Picture two tiny metal spheres separated by a distance d and connected by a fine wire. At time t the charge on the upper sphere is q(t), and the charge on the lower sphere is −q(t). Suppose that we drive the charge back and forth through the wire, from one end to the other, at an angular frequency ω (Figure below)

(a) Using the dipole approximation calculate the Electric field and magnetic field at some point in the wave zone. And then calculate Poynting vector.

(b) What is the time average of the angular distribution and the total power radiated by this dipole in SI units?

(c) Find the radiation resistance of the wire joining the two ends of the dipole. (This is the resistance that would give the same average power loss to heat as the oscillating dipole in fact puts out in the form of radiation.) Show thatR = 790(d/λ) 2 Ohm, where λ is the wavelength of the radiation (Use 0 = 8.854×10−12Fm−1 , c = 3×108ms−1 ). For the wires in an ordinary radio (say, d = 5 cm), should you worry about the radiative contribution to the total resistance?

6. There is a magnetic field given as below: B~ = B0 ˆk , when x ≥ 0 = 0 , when x < 0>

A particle of mass m and charge q moving with a uniform velocity ~v = v0 ˆi enters the magnetic field zone (x ≥ 0) from the non magnetic field zone (x < 0 xss=removed>

(a) How much total energy ED is radiated by the particle as dipole radiation?

(b) How much total energy EQ is radiated by the particle as quadrupole radiation? What is the ratio EQ ED ?

(c) Is there any magnetic dipole radiation? Give appropriate justification

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