### Moving Charges and Magnetic Fields

Still struggling with grant writing season, so another post which has resulted in my random musings about the Universe (which actually happens quite a lot).

In second semester, I am teaching electricity and magnetism to our First Year Advanced Class. I really enjoy teaching this class as the kids are on the ball and can ask some deep and meaningful questions.

But the course is not ideal. Why? Because we teach from a textbook and the problem is that virtually all modern text books are almost the same. Science is trotted out in an almost historical progression. But it does not have to be taught that way.

In fact, it would be great if we could start with Hamiltonian and Lagrangian approaches, and derive physics from a top down approach. We're told that it's mathematically too challenging, but it really isn't. In fact, I would start with a book like The Theoretical Minimum, not some multicoloured compendium of physics.

We have to work with what we have!

One of the key concepts that we have to get across is that electricity and magnetism are not really two separate things, but are actually two sides of the same coin. And, in the world of classical physics, it was the outstanding work of James Clerk Maxwell who provided the mathematical framework that broad them together. Maxwell gave us his famous equations that underpin electro-magnetism.
Again, being the advanced class, we can go beyond this and look at the work that came after Maxwell, and that was the work by Albert Einstein, especially Special Theory of Relativity.

The wonderful thing about special relativity is that the mix of electric and magnetic fields depends upon the motion of an observer. One person sees a particular configuration of electric and magnetic fields, and another observer, moving relative to the first, will see a different mix of electric and magnetic fields.

This is nice to say, but what does it actually mean? Can we do anything with it to help understand electricity and magnetism a little more? I think so.

In this course (and EM courses in general) we spend a lot of time calculating the electric field of a static charge distribution. For this, we use the rather marvellous Gauss's law, that relates the electric field distribution to the underlying charges.
I've written about this wonderful law before, and should how you can use symmetries (i.e. nice simple shapes like spheres, boxes and cylinders) to calculate the electric field.

Then we come to the sources of magnetic field. And things, well, get messy. There are some rules we can use, but it's, well, as I said, messy.

We know that magnetic fields are due to moving charges, but what's the magnetic field of a lonely little charge moving on its own? Looks something like this
Where does this come from? And how do you calculate it? Is there an easier way?

And the answer is yes! The kids have done a touch of special relativity at high school and (without really knowing it in detail) have seen the Lorentz transformations. Now, introductory lessons on special relativity often harp on about swimming back and forth across rivers, or something like that, and have a merry dance before getting to the point. And the transforms are presented as a way to map coordinators from one observer to another, but they are much more powerful than that.

You can use them to transform vectors from one observers viewpoint to another. Including electric and magnetic fields. And these are simple algebra.

where we also have the famous Lorentz factor. So, what does this set of equations tell us? Well, if we have an observer who sees a particular electric field (Ex,Ey,Ez), and magnetic field (Bx,By,Bz), then an observer moving with a velocity v (in the x-direction) with see the electric and magnetic fields with the primed components.

Now, we know that the electric field of an isolated charge at rest is. We can use Gauss's law and it tells us that the field is spherically symmetrical and looks like this
The field drops off in strength with the square of the distance. What would be the electric and magnetic fields if this charge was trundling past us at a velocity v? Easy, we just use the Lorentz transforms to tell us. We know exactly what the electric field looks like of the charge at rest, and we know that, at rest, there is no magnetic field.

Being as lazy as I am, I didn't want to calculate anything by hand, so I chucked it into MATLAB, a mathematical environment that many students have access too. I'm not going to be an apologist for MATLAB's default graphics style (which I think sucks - but there are, with a bit of work, solutions).

Anyway, here's a charge at rest. The blue arrows are the electric field. No magnetic field, remember!
So, top left is a view along the x-axis, then y, then z, then a 3-D view. Cool!

Now, what does this charge look like if it is moving relative to me? Throw it into the Lorentz transforms, and voila!

MAGNETIC FIELDS!!! The charge is moving along the x-axis with respect to me, and when we look along x we can see that the magnetic fields wrap around the direction of motion (remember your right hand grip rule kids!).

That was for a velocity of 10% the speed of light. Let's what it up to 99.999%
The electric field gets distorted also!

Students also use Gauss's law to calculate the electric field of an infinitely long line of charge. Now the strength of the field drops off as the inverse of the distance from the line of charge.

Now, let's consider an observer moving at a velocity relative to the line of charge.
Excellent! Similar to what we saw before, and what we would expect. The magnetic field curls around the moving line of charge (which, of course, is simply an electric current).

Didn't we know that, you say? Yes, but I think this is more powerful, not only to reveal the relativistic relationship between the electric and magnetic fields, but also once you have written the few lines of algebraic code in MATLAB (or python or whatever the kids are using these days) you can ask about more complicated situations. You can play with physics (which, IMHO, is how you really understand it).

So, to round off, what's the magnetic field of a perpendicular infinite line of charge moving with respect to you. I am sure you could, with a bit of work, calculate it with usual mathematical approaches, but let's just take a look.

Here's at rest
A bit like further up, but now pointing along a different axis.

Before we add velocity, you physicists and budding physicists make a prediction! Here goes! A tenth the velocity of light and we get
I dunno if we were expecting that! Remember, top left is looking along the x-axis, along the direction of motion. So we have created some magnetic structure. Just not the simple structure we normally see!

And now at 99.99% we get
And, of course, I could play with lots of other geometries, like what happens if you move a ring of charge etc. But let's not get too excited, and come back to that another day.

1. I do very much agree with your post - and I think that writing down the Maxwell-equations like shown on the t-shirt misses an important point: *after* choosing a frame of reference they assume this shape, while it'd be much better to start off with a covariant formulation which is valid in any frame. in addition, I find that writing down the Maxwell-equations like this masks a bit their physical meaning.

2. "The wonderful thing about special relativity is that the mix of electric and magnetic fields depends upon the motion of an observer. One person sees a particular configuration of electric and magnetic fields, and another observer, moving relative to the first, will see a different mix of electric and magnetic fields."

What about GR? A charge at rest on my desk, in a gravitational field, does not radiate. An accelerated charge radiates. Does this violate the equivalence principle? Discuss.

3. Great post I was checking continuously this blog and I’m impressed! Very helpful info specially the last part :) I handle such info much. I was looking for this particular information for a long time. Thanks and good luck :
Packers And Movers Hulimavu bengaluru
Packers Movers Bangalore
Packers And Movers Electronics City bengaluru
Packers And Movers Bangalore guarantee to convey all your stuff from way to entryway. We don't call ourselves the #best #packers and #movers in #Bangalore simply like that. It is our administrations that backing our case.

4. I must say you had done a great job, I appreciate all your efforts. Thanks a lot for your writings. packers and movers marathahalli packers and movers Bangalore marathahalli

5. 3d bioprinting = Immortality = go to stars

6. Packers and Movers Indore - Call 09303355424, Local Household Shifting in Indore, Domestic Home Relocation from Indore. International Home Relocation from Indore, Office Shifting within Indore, Car and Bike Transportation from Indore, Safe Packing and Moving Services,
Packers and Movers Indore
Packers and Movers Gurgaon
Packers and Movers Kolkata
Packers and Movers Mumbai
Packers and Movers Delhi
Packers and Movers Jaipur
Packers and Movers Raipur
Packers and Movers Pune
Packers and Movers Indore Blog

7. Hello! The article was helpful, I learned many new things on Electricity and magnetism and I also subscribed your website to get updated.

8. My name is Neha Gupta. I run my independent Mumbai Escorts Service. I am an independent Mumbai escort girl. My service charge is low according the market price. My service is super, because I know very well the personal need of my each customer. I am comfortable to provide my Mumbai Escorts Service at your home or in Hotel. 24X7 I am available. Visit- http://www.nehagupta.biz/ Call- +91 9OO4458359
Visit the Links Below and Have a Look At My Various Mumbai Escort Services-

Mumbai Escorts
escort-girl-in-mumbai

Escort Service In Mumbai
Mumbai Escort Service
Andheri Escorts Service
Mumbai Call Girls

9. I Saw Your Website Carefully. Thanks For Create Such Kind Of Informative Website. Your All Content Is Relevant To Your Subject. I Say Keep It Continue, Because Your Website And Subject Is Meaningful For The Users. I Have Recommended Your Website With My Friends Also. I Am Ritu Sharma. I Am An Independent Mumbai Escort Girl. I Deal In Mumbai Escorts service. I Am Very Hot And Sexy. Being A Mumbai Escort Girl I Have Five Years’ Experience. That’s Why I Understand The Each Personal Needs Of My Every Client. My Service Charge Is Low And Service Is Super. You Can Avail My VIP Mumbai Escorts Service At Your Home Or In Hotel. I Am Comfortable To Provide Mumbai Escorts Service In Local And Outside Also. Visit www.escortservicemumbai.net Call +91- 9OO4OO9481
Visit The Links Below And Have A Look At My Various Mumbai Escort Services-
Mumbai Escorts
Mumbai Call Girls
Mumbai Escorts In Vile
Parle

Bandra Escorts
Juhu Escorts
Andheri Escorts

Mumbai Escorts In Colaba

10. My Name Is Priya Singh. I Run My Own Mumbai Escorts Service. I Am An Independent Mumbai Escort Girl. I Am Beautiful And Hot. My Service Charge Is Low And Service Is Super. Being Professional I Have Seven Years’ Experience As An Escort Girl. So I Understand And Feel The Real Needs And Requirement Of My Each Client. You Can Taste Me Any Time. According The Convenience You Can Avail My VIP Escort Service At Your Home Or In Hotel Also. To Book My VIP Mumbai Escort Service Call +91 9987215552. Visit http://www.escortagencyinmumbai.com/

Mumbai Call Girls
Mumbai Escorts In Vile Parle
Bandra Escorts
Juhu Escorts
Andheri Escorts
Escort Service In Colaba

11. My name is Arpita Jain. I run my independent Mumbai Escorts Service. I am an independent Mumbai escort girl. My service charge is low according the current market price. My service is super, because I know very well the personal requirement of my each customer. I am comfortable to provide my Mumbai Escorts Service at your home or in Hotel. Visit- http://www.arpitajain.org/ Call- +91 9OO4458359
Visit The Links Below And Have A Look At My Various Mumbai Escort Services-

Mumbai Escorts Service
Mumbai Escort Girls
Juhu Escorts

Escort Service In Mumbai

12. an awesome blog thanks a lot for giving me this great opportunity to write on this.please visit our website I assure you that it will be very useful for you @ Packers and Movers mumbai

13. In this competitive world, people require moving from one position to another to meet employ & domestic needs. Present is constant rise in the Indian country, lead to involve the mobility of people. Whether it is a house or office, present is no impedance in the repositioning process due to the strength of political and economic borders. While relocate, a lot of equipment come to mind of the public, regarding the protection and care of their possessions, we, at maxway Packers and Movers Lucknow, offer you the services and solution for what you desire.

With Regards,
Maxway Packers and Movers Lucknow

http://maxwaypackersandmoverslucknow.co.in/

14. Looking for professional and licensed verified Packers and Movers at an affordable cost?
Your search ends here... Hire the most licensed and verified packers and movers at an affordable cost and save money, compare the quotes with the top 3 competitors and choose the best one!
100% Licensed & verified Transporters in India

Packers and Movers India | Licensed Transporters | Hassle-free relocation
Call now : 8886663106

Packers and Movers in Bangalore

15. Bangalore escort services are largely here to give you an astonishing encounter. The most electrifying hours with intriguing Bangalore escorts are on your way.
Bangalore Escorts
Bangalore Model Escort
Air Hostess Escorts in Bangalore
Elite escorts Bangalore

16. Packers and Movers Hyderabad Give Certified and Verified Service Providers, Cheap and Best ###Office Relocation Charges, ***Home Shifting, ✔✔✔Goods Insurance worth Rs. 10,000, Assurance for Local and Domestic House Shifting. Safe and Reliable Household Shifting Services in Hyderabad with Reasonable Packers and Movers Price Quotation @ Packers And Movers Hyderabad

17. We have calls available in the best of categories, according to our need, we are waiting for you, erotic and beautiful call girls in Gurgaon escort service, you can come on our website and contact us 24x7 hours anytime in Gurgaon. I look forward.
Escort In Gurgaon
Gurgaon Escort Service
Escort Service In Gurgaon
Female Escort In Gurgaon
Independent Escort In Gurgaon
Gurgaon Escort With Whatsapp Number
Gurgaon Call Girls
Girls In Gurgaon
Escort in Gurgaon

18. Escort girls in Delhi make independent Brahma; Delhi call girls are always ready to serve you for interested people. Call girls in Delhi have been placed in many categories. Surely we are available according to you, you can come on our website and contact us. Female Escort In Delhi

19. Diverse Karachi Call Girls Services The posting of the Call Ladies administrations you may get from these kinds of call ladies is unlimited,Escorts at Karachi You'll get all the advantages out of our Karachi Call Girls.

20. Today Best Choice is expecting a phenomenal activity in the satisfaction of the customer. That is the clarification in Islamabad Escorts Agency we are giving the most loosened up choice to pick an appealing call girls in islamabad . We have cutest call youngsters and hot dears who draw in your whole night in your room. Are you arranged to pick a crazy and alluring newborn child in Pakistan Escorts Agency how about we do it magnificently

21. A significant number Of The Escorts in Karachi Friendly Call Girl Services Are You Interesting Local To Vocal in Pakistan Hiring
Proficient Worker Manager in Hotel Rooms With COvid Free Environments in Karachi

22. Lahore Escorts offers exceptionally taught model from Lahore and best Escorts administration in Lahore through a portion of its sizzling Independent escorts in Lahore Call Girls who are accessible to clients 24*7.

23. You want to move your office or home appliances? Then you are at the right place. Use the best House Movers to move good things safe and sound.

24. Thanks for sharing this useful information! This is really interesting information to read. Best high-quality Packing And Moving services at affordable prices within your budget.
@ Packers and Movers Bangalore
@ MovingNow Packers and Movers Blog
@ MovingNow Packers and Movers Blog

25. Pioneer Cranes & Elevators is one of the Best EOT Cranes Manufacturers. All Cranes equipment is manufactured in peace to International Standards and is of high quality and expertise. The team here is very professional and skillful in providing the customer with the best services. Excellent Quality customer service is a main part of our business that way we provide 24x7 customer service. Moreover, entire range of EOT Cranes, JIB Cranes, Overhead Crane, Single Girder EOT Crane, Wire Rope Hoists is tested on various parameters of quality in demand to deliver perfect product at clients' end.

27. Hello there,
Vansh Packers and Movers Agra is regarded as one of the Best Packers and Movers in Agra, well-known for their moving services as well as quality and budget packing. Best Packers and Movers Firozabad which provide door to door services.

More Citywise Packers and Movers Service:-
Packers and Movers Hathras
Packers and Movers Mathura
Packers and Movers Aligarh
Packers and Movers Etawah
Packers and Movers Etah

Why Vansh Packers and Movers Agra?
✓ISO & IBA Verified Packers and Movers in Agra
✓Safe & Secure Reliable House Shifting Services
✓Compare our Reasonable Rates
✓Hassle Free Packing/Unpacking Shifting Services
✓No Hidden Cost & No Surprises
✓We Are Contactable & Responsible

Name: Mr. Hotam Singh
Mobile: +91 8006720005 (WhatsApp & Call)
https://www.packersandmoversagra.com

28. Deep Learning Projects assist final year students with improving your applied Deep Learning skills rapidly while allowing you to investigate an intriguing point. Furthermore, you can include Deep Learning projects for final year into your portfolio, making it simpler to get a vocation, discover cool profession openings, and Deep Learning Projects for Final Year even arrange a more significant compensation.

Python Training in Chennai Project Centers in Chennai

29. Tanks for sharing if you are looking best house movers in Kitchener services in canada than join Trans Movers services

30. This Blog Is really informative for us. Thanks For sharing this blog.
Hii this is my first time visiting this web page this blog is really informative for me. thanks for sharing. keep posting.
Visit for moving packing in Patna:-

packers and movers in patna
movers packers in patna

31. Excellent article!!! Some crucial considerations for anyone looking for a moving company to assist them with their relocation. There are other aspects to consider, including safety, speed, and cost. One crucial point you made above is that you should check the company's credentials Packers and Movers in Noida, which is the most significant. Thank you for these suggestions.

32. Hello Bloggers,

I really love your post. It’s very informative to us. Keep sharing your post.
We offered service like:
Vansh Packers and Movers Agra provides Best Packers and Movers in Firozabad with fully insured. Services Offered: Office Shifting, Household Shifting, Packing & Unpacking, Car & Bike Transport.

More Cities:
Vansh Packers and Movers in Mathura
Vansh Packers and Movers in Aligarh
Vansh Packers and Movers in Etawah

Why Vansh Packers and Movers Agra?
✓ISO & IBA Verified Packers and Movers
✓Safe and Secure Reliable House Shifting
✓Compare our Reasonable Rates
✓Hassle Free Packing/Unpacking
✓No Hidden Cost & No Surprises
✓We Are Contactable & Responsible

Name: Mr. Hotam Singh
Mobile: +91 8006720005 (WhatsApp & Call)
https://www.packersandmoversagra.com

33. Hi Blogger,
Vansh Packers and Movers Agra provide door to door service to Bharatpur with experience workers. Contact for cheap Packers and Movers Bharatpur. Household Shifting, Packing and Unpacking in Bharatpur. Call and WhatsApp: - +91 8006720005 and Visit website:- https://www.packersandmoversagra.com/bharatpur.html

Thanks
Hotam

34. Vansh Packers and Movers Agra offers the best Packers and Movers Mathura services, including household shifting, Office Shifting, door-to-door services, and complete insurance. Call Us 08006720005 for free quotation Packers and Movers. visit site: www.packersandmoversagra.com for more details.

35. Your article is fantastic. It's quite useful to us. Continue to share your article.
Vansh Packers and Movers Agra offers completely insured Best Packers and Movers Mathura services. Office Shifting, Household Shifting, Packing & Unpacking, Car & Bike Transport are some of the services provided. visit site: www.packersandmoversagra.com for more details.

36. Awesome post, you have shared an informative article about test and tag services sydney . Your method of writing is great. Keep it up.