Radiating dipoles

Problem:

An electric dipole p0 vibrates with frequency ω.
(a)  How will the total radiated power change if the frequency is doubled?
(b)  Find the ratio of the differential power in the direction of θ = 45o to the dipole's axis to the differential power in the direction perpendicular to the axis. 

 

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Solution:

Problem:

An electric dipole p(t) = p0(z/z)cosωt is placed at the origin. 
(a)  Determine the radiation fields on the z-axis.
(b)  Determine the radiation fields for a direction normal to the z-axis.
(c)  How does the intensity of the radiation vary as a function of θ?
(d)  How does the total power radiated vary as a function of ω?

Solution:

Problem:

A thin spherical shell of radius a has a surface potential Φ(a,θ) = Φ0 cosθ, where θ is the usual polar angle relative to the z-axis, and Φ0 is a constant. 
(a)  Find the electric potential Φ(r,θ) everywhere outside the shell.  Calculate the equivalent electric dipole moment p that produces this potential Φ.
(b)  Assume that previously calculated dipole moment acquires a cos(ωt) time dependence.  Calculate the magnetic field B and the electric field E in the radiation zone.
(c)  Find the time-averaged power d<P>/dΩ radiated per unit solid angle.  Express your result in terms of ω, θ, a, Φ0, ε0 and c.
(d)  Find the total power <P> radiated.

Solution:

Problem:

An electric dipole p0 makes an angle of θ degrees with respect to the z-axis as it rotates about the z-axis with angular speed ω.  Find its initial rate of energy loss.

Solution:

Problem:

A small current loop of area A and N turns carries a sinusoidally varying current I(t) = I0sin(ωt).
(a)  Find the magnetic moment of the loop.
(b)  Find the average power radiated by the loop.

Solution:

Problem:

A rotating neutron star slows down due to dipole radiation.   Assume that a neutron star has a surface magnetic field strength of 100 Mega Tesla at the poles, a radius of 10 km, and nuclear density.  The neutron star's magnetic axis is tilted by 45 degrees relative to the axis of rotation of the neutron star.  How long will it take for the neutron star's rotation rate to decrease by a factor of 3 if the initial rotation rate is 105 revolutions per second?

Solution: