### 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. 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

4. 3d bioprinting = Immortality = go to stars

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

6. 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/

7. 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

8. 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.

9. 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

10. 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.

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

12. 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.

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

14. This comment has been removed by the author.

15. This article is a gem! Thanks for sharing such informative content, I appreciate Superman Junk and Moving's commitment to the environment. Their waste management and garbage dump removal services made us feel good about our move. It's more than a moving service; it's a sustainable solution."

16. In today’s digital age, buying YouTube subscribers through Paytm emerges as a smart, strategic move for content creators looking to elevate their channel's visibility and influence. This seamless strategy empowers creators by rapidly boosting their subscriber counts, a crucial factor for gaining traction and standing out in a crowded content landscape. Leveraging Paytm’s trusted and secure payment platform, creators can confidently invest in their growth, ensuring that every transaction is both safe and straightforward. This approach is not only efficient but also cost-effective, making it accessible to creators at all levels, from novices to veterans. Increased subscriber numbers naturally lead to improved channel authority, fostering organic viewer engagement and expanding audience reach. By tapping into this innovative service, YouTube content creators can set a strong foundation for success, fostering a vibrant community around their content and driving forward their digital presence.

17. LASIK eye surgery in Delhi represents the pinnacle of corrective vision procedures, offering a blend of innovative technology and expert surgical skill. In this bustling metropolis, state-of-the-art clinics provide a gateway to improved sight, utilizing advanced laser technologies to reshape the cornea with precision and care. Patients are treated to a holistic experience that begins with a thorough consultation, ensuring each individual's needs and concerns are addressed. The surgeons in Delhi are not just professionals; they are visionaries, whose expertise and compassionate approach make the LASIK process seamless and stress-free. With competitive pricing and a focus on safety and efficacy, these surgeries promise a significant enhancement in quality of life. The success stories echoing from satisfied individuals stand testament to the life-changing impact of LASIK surgery in Delhi. It is an option for those seeking to liberate themselves from the constraints of glasses or contact lenses, offering a clear vision of the future.
https://www.visualaidscentre.com/

19. Cheap web hosting services in India present a cost-effective gateway for budding entrepreneurs and small businesses to carve out their digital space. These services are designed with the budget-conscious user in mind, offering a balance between affordability and functionality without compromising on essential features. Customers can expect a suite of services including reliable uptime, sufficient storage and bandwidth, comprehensive security measures, and support for popular CMS platforms. Additionally, these hosting packages often come with intuitive control panels, simplifying website management for users of all expertise levels. The goal is to democratize access to the web, allowing even the smallest of ventures to establish a robust online presence and connect with a global audience.
https://onohosting.com/

20. Dynamic Health Staff is a pioneering recruitment agency committed to connecting healthcare professionals with leading healthcare facilities around the globe. Specializing in a wide array of medical fields, they offer personalized recruitment services to ensure both employers and candidates find their perfect match. With a focus on empowering professionals in their career growth, they provide extensive support throughout the entire recruitment process, including licensing and immigration advice. Leveraging a vast network of healthcare institutions, Dynamic Health Staff is dedicated to ensuring the highest standards in healthcare staffing. Their expertise and commitment to excellence make them an invaluable partner for healthcare professionals seeking to elevate their careers and healthcare facilities aiming to strengthen their teams with top-tier talent.
https://dynamichealthstaff.com/

21. Choosing the right breast specialist doctor in Ahmedabad is crucial for addressing concerns related to breast health. These specialists are highly trained in diagnosing and treating conditions such as breast cancer, benign breast diseases, and other abnormalities. They employ advanced diagnostic techniques, including mammography, ultrasound, and MRI, to ensure accurate detection and assessment. With a compassionate understanding of their patients' needs, breast specialists in Ahmedabad provide personalised treatment plans which may include surgery, chemotherapy, radiation, and hormonal therapy. Many professionals in this field also focus on breast reconstruction and aesthetic surgery to support women through recovery and beyond. Consulting an experienced breast specialist ensures comprehensive care and peace of mind.
https://drvirajlavingia.com/

22. A Breast Specialist Doctor in Mumbai offers comprehensive care for individuals facing various breast health issues, including breast cancer, benign lumps, and other abnormalities. These experts are proficient in utilizing advanced diagnostic tools such as mammograms, ultrasounds, and MRIs to ensure early and accurate detection of breast conditions. With a patient-centered approach, they create personalized treatment plans that may involve surgical and non-surgical options, including lumpectomies, mastectomies, chemotherapy, and radiation therapy. The specialists in Mumbai also focus on the psychological well-being of their patients, providing emotional support throughout the treatment journey. Many are skilled in performing breast reconstruction and aesthetic surgeries, aiding in both the physical and emotional recovery. Consulting a Breast Specialist Doctor in Mumbai ensures access to state-of-the-art medical care and holistic support for optimal health outcomes.
https://drnitanair.com/

23. A Breast Specialist Doctor in Gurgaon is dedicated to providing expert care for breast health issues, encompassing both benign and malignant conditions. These specialists are skilled in advanced diagnostic techniques such as mammograms, ultrasounds, MRIs, and biopsies to ensure accurate and early detection. They offer a range of treatment options, including surgical interventions like lumpectomies and mastectomies, as well as non-surgical treatments such as chemotherapy and radiation therapy. The focus is on holistic care, integrating emotional and psychological support throughout the patient's journey. By collaborating with a multidisciplinary team of oncologists, radiologists, and reconstructive surgeons, they ensure comprehensive and individualized treatment plans. Consulting a Breast Specialist Doctor in Gurgaon ensures access to state-of-the-art medical facilities and compassionate care, aiming to improve both health outcomes and quality of life.
https://www.breastoncosurgery.com/

24. A Breast Specialist Doctor in Pune is dedicated to providing top-tier care for various breast-related conditions. These experts are highly skilled in diagnosing and treating breast cancer, benign lumps, and other breast diseases using state-of-the-art imaging techniques like mammography, ultrasound, and MRI. They craft personalized treatment plans that may include surgery, chemotherapy, radiation, and hormonal therapies. Beyond medical treatment, they offer emotional support and guidance to patients throughout their journey. With a patient-centred approach, these specialists ensure holistic care that addresses both physical and emotional needs. Their commitment to excellence and access to advanced medical facilities contribute significantly to improving patient outcomes and quality of life.
https://www.drshonanagbreastcancer.in/

25. Minoxidil is a widely recognized topical treatment for hair loss, particularly effective in addressing androgenetic alopecia in both men and women. Applied directly to the scalp, it works by dilating blood vessels and revitalizing shrunken hair follicles, thereby promoting a longer growth phase for hair. Consistent use, typically once or twice daily, can yield noticeable improvements in hair density and health over several months. Clinical studies support Minoxidil's ability to stimulate new hair growth and slow down hair thinning. While it is generally well-tolerated, some users may experience mild side effects such as scalp irritation or dryness. Consultation with a healthcare provider is recommended to determine the appropriate usage and dosage tailored to individual needs.
https://generichealth.com.au/minoxidil-3/

26. Silk Eye Surgery is an innovative vision correction procedure available in Delhi, designed to treat refractive errors such as myopia, hyperopia, and astigmatism with ultimate precision. This advanced technique employs sophisticated laser technology to gently reshape the cornea, providing enhanced visual clarity while reducing dependency on glasses or contact lenses. Renowned for its minimally invasive approach, Silk Eye Surgery ensures a swift recovery and minimal discomfort. Leading ophthalmologists in Delhi prioritize patient safety and comfort, offering personalized pre-surgery consultations to devise bespoke treatment plans. The high success rates and overwhelmingly positive patient testimonials reflect the procedure's effectiveness. Notable eye care centers in Delhi are celebrated for their exceptional patient care and meticulous post-surgery follow-up. For top-quality Silk Eye Surgery in Delhi, consult the city's expert eye specialists.
https://medium.com/@pojagupta

27. Smile Pro Eye Surgery is a state-of-the-art, minimally invasive vision correction procedure that uses advanced femtosecond laser technology to treat myopia and astigmatism. By making a small, precise incision in the cornea, it significantly reduces recovery time and post-operative discomfort compared to traditional methods. The procedure typically takes about 15 minutes per eye, with most patients experiencing noticeable improvements in their vision almost immediately. Performed in top-tier eye clinics in Delhi, Smile Pro combines cutting-edge technology with the expertise of skilled ophthalmologists, ensuring high success rates and patient satisfaction. It is a reliable and efficient solution for those looking to achieve clear, perfect vision without the need for glasses or contact lenses.

28. Our mobile app development company in Delhi stands at the forefront of crafting high-quality, tailor-made applications for both Android and iOS platforms. Utilizing advanced technologies, we develop secure, scalable, and user-friendly mobile solutions. We cater to a diverse range of industries including healthcare, finance, and e-commerce, ensuring each app meets specific business objectives. Our skilled team follows agile methodologies for seamless project management from inception to deployment. Post-launch, we provide robust support and maintenance services. Partner with us to elevate your innovative ideas into powerful, competitive mobile applications.
https://olycoder.com/

29. Low-risk, high-return investments in India attract conservative investors by providing stable returns while minimizing exposure to market volatility. The Public Provident Fund (PPF) is a reliable option, known for its government backing, attractive interest rates, and tax benefits. Fixed Deposits (FDs) offered by reputable banks and NBFCs guarantee fixed returns over specified terms. National Savings Certificates (NSC) deliver secure and consistent returns with tax advantages. Debt Mutual Funds allocate investments across high-quality government and corporate bonds, balancing risk and reward. Sovereign Gold Bonds (SGBs) offer returns linked to gold prices, acting as a safeguard against inflation. RBI Floating Rate Savings Bonds adjust interest rates with inflation, preserving purchasing power.
https://www.perannum.money/

30. House Clearance Edinburgh provides compassionate, efficient, and environmentally-responsible clearing services for homes and offices. Our dedicated team is experienced in handling various situations, including bereavement clearances, downsizing, and decluttering projects. We approach every job with respect, ensuring personal belongings are treated with care throughout the process. With a focus on sustainability, we prioritize recycling and donating usable items to minimize waste and support the community. Our transparent pricing model ensures you’re aware of all costs upfront, with no hidden fees. Flexible scheduling options allow us to work around your availability for maximum convenience. At House Clearance Edinburgh, our commitment to customer satisfaction drives everything we do. Contact us for a free, no-obligation quote and start reclaiming your space today!
https://eh1-edinburghremovals.co.uk/house-clearance/