Tuesday, August 4, 2026

Solutions To Single Particle Dynamics and Hertizian Dipole Problems

 1)Given the situation shown below for a particle in plasma:


And: v o = v xo x^ + v yo y^ (for initial velocity)

Solve the equation of motion: F = q(E + v X B)

Solution:

m dv/dt = q(E + v X B)

v x   =   qB (v y)/m   +  q E x /m

v y  =  - qB (v x)/m  

Or: 

v y  =  - qB/m [qB (v y)/m   +  q E x /m]

=  - W e 2 (v y)   -  W e 2 (E x) / B 

v x   =   - W  e 2 (v x) -  W  e 2 (E x ) / B 

 

2) (a)If the perpendicular velocity component ( v) is 10 m/s for an electron in a plasma, find its Larmor radius, gyration energy and its gyro-period.

Solution.

Larmor radius:  r  = m/ q [v / B] =  v/ (qB/ m e) = v/ e

 r = (10 5  m/s) / 1.7 x 10 7  /s   =     0.0056 m or:  0.56 cm

Gyro-period: T = 2 p  / e  =   2 p / 1.7 x 10 7  /s   =  3.5 x 10 -7  s

Gyration energy E = m e  (v)2/ 2 =  
  (9.1 x 10 -31  kg) (10 5  m/s) 2  / 2

 E =    4.5 x 10 -21  J  

 = 0.028 eV

(b) Find the guiding center positions for the electron referenced in (a) if t = T/2.

Solution:

  The gyro-period is: T = 2 p  / Ω  

Then: T/2 =   2 p  /2 Ω  =  p / Ω  


Bear in mind the gyration energy:

E  =
m m  B = m/2 (E/B) 2,

Guiding center positions:

 x – xo =   r sin (e t)   =  (0.0056 m)  sin (e   p  / e  ) = (0.0056 m) sin p

=  (0.0056 m) 0 = 0

y – yo =    r  cos (e   p  / e  )   =    

(0.0056 m) cos p   =   - 0.0056 m


The Hertzian Dipole Problem Solution:

The Problem:

Find the total average power (Prad) radiated by a Hertzian dipole over a closed sphere of radius r.   (Take  ho as the impedance of free space, with I  the steady current and k  =  w /c  the wave number.

The Solution





Assessing The Two-Year Mark Efficacy Of The Anti-Androgen Treatment For My Metastatic Prostate Cancer

 

                                   Firmagon 'depot' formed after injection

It hasn't been easy but after 24 months of hot flashes, sleep turmoil, back aches and general low energy (plus cognitive 'fuzz' and memory lapses) the androgen deprivation therapy with Firmagon now shows some decent progress.  At least according to the latest MRI back in May.

Namely:

- Elimination of the metastases in the lymph nodes which were evident two years ago, and

-Reduction of the number of metastastic adenocarcinomas in the prostate gland to one - down from two. 

 The treatment of choice was one Firmagon injection in the belly each month, the graphic at top showing the needle's entry and the 'depot' or the mass of the chemical left behind. 

Firmagon, far less any other androgen deprivation treatment (ADT), was not my choice as I will admit. But a PSMA scan in March, 2024 showing the spread of lesions (with greater SUV intensity) to pelvic region and lymph nodes changed that. (Intensity of  uptake of radionuclide by cancer cells is defined by the SUV or standardized uptake value.  

Where: SUV = C(T)/[injection dose (MBq)/patient's weight (kg)] 

So the sole remaining option was to go on ADT which consequences- side effects are summarized in the table below:


I had originally hoped to start Orgovyx, the newest (and ‘greatest’) ADT medication which had the benefit of: a) being taken in pill form as opposed to regular injections - which usually leave massive swelling at the injection site, and b) fewer myocardial infarctions ( by about 43%) .  But the path to support was scuttled because the company terminated its lower cost  “bridge support.  Simply put, I was not prepared to cough up $2,500 a month for this solution. 

Both Firmagon and Orgovyx work by blocking the pituitary gland from making hormones called luteinizing hormone and follicle-stimulating hormone, thereby reducing the amount of testosterone the testicles are able to make.   With testosterone reduction the cancer's fuel intake is lowered and hence there is less havoc caused by tumor growth - including metastasis.  

Let me quickly add here the ADT regimen is rarely used for early prostate cancer.  But there are more sites now that will, when PSMA scans show suspicious SUV values, deliver radiation - usually SBRT ( Stereotactic body radiotherapy  ) with ADT.  I also had the option but declined the radiation given I already had high dose brachytherapy 12 years ago,  

 Of course, my urologist has been adamant I need to go further to drive the "T" down to 0 from the already castrate level of 10 ng/dl.  This is to ensure the remaining lesion-tumor, some 0.8 cm across, does not "escape the capsule", i.e break through and grow into the rectum.  To that end he is proposing the addition of another ADT drug (Nubequa) as well as a form of immunotherapy for metastatic prostate cancer.  This combined regimen is currently scheduled to begin this month, pending Medicare approval.

The latter involves use of 'provenge' - a personalized cellular vaccine used for asymptomatic or minimally symptomatic metastatic castration-resistant prostate cancer. It involves extracting white blood cells, exposing them to a prostate protein in a lab, and infusing them back to stimulate an immune response.  I've been informed the worst that can happen in terms of side effects are symptoms mimicking a bad flu during infusion.   Rashes and auto-immune inflammation in healthy organs are also possible.

As for the Nubequa, the possible serious side effects include:

-Weight gain and high blood sugar levels

- Rash

- Increased risk of seizure or momentary loss of consciousness

 - Potential blockage of the arteries in the heart leading to death

Of course, given a calcium scan back in 2025,

Brane Space: The Calcium Scan Score: A Wakeup Call That Low Cholesterol Is No Indicator For Avoiding Heart Attacks

 showed a 57 % blockage in the right coronary artery, that is a cause for concern. And even more reason I want to opt for the lower dosage

Look, I ain't looking forward to any of this, but if I can get a decent cancer remission it will be worth it. My one demand will be to start the Nubequa at the lower dose (300 mg twice a day) as opposed to the higher, though I know the cancer specialist always want to go for the 'kill shot'.  Me, I'm ok with a half shot and not having everything fried that still works.


See Also:

Prostate Cancer and Androgen Deprivation Therapy (youtube.com)

  • And:
       
Androgen Deprivation Therapy: What Metastatic Prostate Cancer Patients (Like Joe Biden) Will Now Face - As I Have (Pt. 1)

       And:


And:

Monday, August 3, 2026

Solution To First Order DE Problem - Eliminating Dependent Variable (Solving For x)

 Solve by finding a general solution: 

py 2 -  x = 0  

Re-arrange to express x alone: 

x=   py 2 

Replace p with dy/dx:

y 2  (dy/dx)  -  x  = 0

Separate variables:

y 2  (dy/dx)  =  x  

Set up for integration:

y 2  ò  dy  =   x ò dx

Yielding:

  y 3 /3  =   x 2 / 2  + c

®

 x 2    =  2 y 3 /3  - 2C

Let  C =  -2c

(New arbitrary constant)

Solving for x to get general soln.

x  =  + Ö  2 y 3 /3  +  c



NY Times' Douthat (& Yale's Moyn) Wrong As Ever About "Older Americans Hoarding The Future"

 

            Douthat (R) and Moyn yuck it up over 'the tyranny of the elderly'



NY Times conservo religionist columnist Ross Douthat is at it again (July 30, 'How Older Americans Hoard The Future') this time uniting with a Yale 'anti-gerontocracy'' prof  (Samuel Moyn) to attack average seniors. Recall Moyn is a guy who'd I'd earlier skewered in a post, e.g.

Yale Econ Prof Claims "Older People" in U.S. Are A "Gerontocracy Stealing From the Young" - Where and How He's Off Base

Douthat barks:

"Forget fascism. Forget socialism. What if the real threat to American democracy is the tyranny of the elderly? "

Adding a bit later:

  "The American welfare state, built by your left-wing forebears, is designed heavily around retirement protections — Medicare, Social Security. These are the third rails of American politics.,"

These two gentlemen, however, might want to get a more informed perspective on who the real 'hoarders' are by checking out the WSJ Review piece ('When One Nanny Isn’t Enough, ‘CEO Parents’ Hire a Whole Team', July 25-26, p. C4 ) wherein we are first introduced to the lead -in:

"One nanny isn’t cutting it anymore. Some parents are spending upward of $250,000 on teams for their children. Luxury services offer potty-training, baby chefs and bike-riding lessons."

Then learn this about one 27-year old multi-millionaire mother:

"As a mother of two, there are certain tasks Christine Landis won’t do. She has never clipped her children’s toenails—too anxious for that, she said, so the nanny handles it. Most nights, the family assistant takes care of bath time. Landis also hands off what she calls “repetitive play,” reading the same book five times or watching her toddler open and close a door for 45 minutes. 

To make this possible, Landis, who lives in San Diego, spends roughly $250,000 a year running a household staff that includes a family assistant who lives on the property, a weekend nanny, a personal chef and a housekeeper."

But hold strain, this one example doesn't even scratch the surface of these uber wealthy hotshots, as we then learn about the 'Pavilion Agency' which helps staff wealthy homes with nannies and other assistants:

"The agency often serves clients who rotate across their 5 to 10 homes, with full staff at each proprerty"

And according to CEO Seth Greenberg (ibid.):

"We are busier than ever. The environment, the economy, it's just producing these millionaires and billionaires."

And it ain't retired oldies but mainly Young Turks who made their millions or billions from a debt-fueled, overpriced AI stock market.

Quoting one "max 3 hour mum" in her early 30s"

"'Cutting sandwiches into special shapes? That was never for me.'

 She relies instead on nannies, cleaners, a chef and a chief of staff.  For families who want full time dedicated staff, a travel nanny costs 4120,000  to $170,000 a year. For a private governess add $200,000."

And then Moyn - and now Douthat- complain about retired oldies using their high medical expenses and Social Security to deprive the young? Gimme a break. Methinks you need to look at these high roller Yuppies as opposed to the elderly exercising a "tyranny of the elderly".  Just sayin'.

See Also:


by Robert Reich | July 30, 2026 - 5:17am | permalink

— from Robert Reich's Substack

Friends,

As Trump’s tariffs and his war with Iran drive up prices of everything from oil to steel, you’d think big American corporations would be screaming bloody hell. Presumably, Trump’s tariffs and war are squeezing their profits just as they’re squeezing the wallets of average Americans.

To the contrary, corporate America is quietly encouraging both Trump’s war and Trump’s tariffs. Why?

One possibility is they’re raking in money off the war as defense contractors and suppliers.

Some surely are, but this can’t account for the acquiescence if not outright support by most big American corporations that have nothing to do with the defense-industrial complex.

» article continues...

And:

by Robert Reich | July 28, 2026 - 5:08am | permalink

— from Robert Reich's Substack

A primer on what's happened — and what to do about it

Friends,

Many of you have asked for a video that sets out simply and clearly how wealth inequality in the United States spiraled out of control, why it’s a huge problem, and what we can (and must) do about it. So here goes:

» article continues...

And:

Youtube videos:


&



Friday, July 31, 2026

Mensa 3D Geometry Tetrahedron Challenge

 

 

Let the 4 vertices of a particular tetrahedron (blue profile)  exist as half the vertices of a unit cube.    

Let the base vertices be one unit away from the fourth (apex) vertex as shown in the diagram above.  

 Given only this information find the height (H) and volume (V) of the tetrahedron.

Hint: The formula for the volume of a tetrahedron is one third the area of the base times the height H.

Advanced E&M Treatment of the Poynting Vector Applied To Hertzian Dipoles

 The energy carried by an electromagnetic (E-M) wave with field intensities E, B is given by the Poynting vector, S:    

S = 1/m [E X B]

The preceding is the most elementary form usually presented in the 2nd semester of Calculus physics.  But in this post I want to treat the more advanced (graduate) level form:

S = 1/m [E X B]   =

 m/ c  {[ P w2 / 4p ]  sin q/ r  cos w2 (t - r/c)]}2    r^

From here we proceed by taking the time average, i.e.:

<S> =    (mo / c  )P2 w4 / 32 p 2  sin 2 q/ r2    r^

Then the total power radiated:      

Ptotal    = mo Po 2 w4 / 32 p 2  c  ò 2p  o  d  ò p   sin 2  sin q dq


And: 

ò p   sin 2  sin q dq  =   ò p   sin q  dq  =  - cos q  + cos3 q / 3 p  o


= - 1 - 1/3  -  (-1 + 1/3)  =  6/3  - 2/3   =  4/3

And:

  ò 2p  o  df  =    2 p

  Þ

Ptotal    =    

  mo Po 2 w4 / 32 p 2  (2 p)  (4/3) =  8/3 [ mo Po 2 w4 / 32 p 2  c]


Ptotal    =    mo Po 2 w4 / 12 p  c

This amounts to the energy radiated from a Hertzian dipole, which is the largest of all types of dipole configuration.  

 Suggested Problem:

Find the total average power (Ptotal av = Prad radiated by a Hertzian dipole over a closed sphere of radius r.   (Take  ho as the impedance of free space, with I  the steady current and k  =  w /c  the wave number.)