# B Can I understand Maxwell's equations on a very basic level?

1. Nov 25, 2016

### Mary curie

Hi guys!!!!
First, I'm a high school student. A senior!
We don't study maxwell equations yet so when I'm doing a research about E=mc^2 and especially ghe energy part I came across electromagnetism and of course these bizarre equations for me !!
I tried to understand them using internet but I failed !!!

2. Nov 25, 2016

### Orodruin

Staff Emeritus
In order to understand Maxwell's equations you need to know vector analysis. This in itself is typically a course of at least several weeks of full time studies at university level. Not to mention the prerequisite courses for that, such as multivariable analysis and linear algebra. Unfortunately, this is time that you will have to invest if you wish to understand Maxwell's equations.

On a more basic level, we could only try to explain the purpose and some features of the solutions.

If you are intent on learning them, I suggest picking up some good university level textbooks.

3. Nov 25, 2016

### Mary curie

You know I start seeing them everywhere and I feel really stupid not knowing them so I think I' m gonna invest in learning them sooner than later !!
Thank you so much !!
Please can u tell me in wich level in uni in physics department I have to look ???

4. Nov 26, 2016

### ZapperZ

Staff Emeritus
Before you learn how to build a house, you need to learn the tools that are needed, and to learn how to use those tools. So it is not just learning the physics, but also learning the mathematics to be able to do the physics.

You need to know partial differential equations and vector calculus before you learn about maxwell equations. There are no shortcuts.

Zz.

5. Nov 26, 2016

### Cutter Ketch

Oh I suppose we can say a little something to give Marie the flavor.

The first Maxwell equation is Gauss' law. We'll see in a minute that you can make loops of electric field, but what Gauss' law says is that if an electric field line has an end, it ends at an electric charge. Or to put it another way, electric charge produces electric field. To be precise Gauss' law says that if you have a closed volume, any closed volume, then the sum of all the electric field normal to the surface of that volume is equal to the sum of the electric charge inside that volume. You could picture some electric field lines. If you pick a volume of space some field lines might penetrate that volume but every line that goes into the volume must also come out. That adds up to zero. The only way to have net field coming out or going in is if there is charge inside.

Gauss' law has lots of implications and uses. Say you have a point charge. Pick a sphere around it. No matter what the size of the sphere the net amount of electric field coming out of the surface is equal to the charge. Since the area of a sphere is 4 pi R^2, the field strength falls off proportional to 1/R^2. That is just one of the many implications of Gauss' law

The second of Maxwell's equations is exactly like Gauss' law for electric field, but this time we are talking about magnetic field. However there are no magnetic monopoles, no "magnetic charge", so this time the sum of the magnetic field over a surface always adds up to zero. In short magnetic field lines only make loops. They can have no ends.

The third of Maxwell's equations is Faraday's law. It says that a time varying magnetic field will induce an electric field, that is to say will cause a force on an electric charge. We know from Gauss' law that the induced electric field must form a loop. Faraday's law is often written in terms of a loop. If you pick a loop, any loop in space, then the sum of the electric field pointing along the rim of the loop is equal to the time rate of change of the sum of magnetic field penetrating the loop.

This idea is used to generate electricity. You coil a wire into a loop (or go around several times to make a lot of loops) then you spin a bar magnet near the loop. The time varying magnetic field from the bar magnet induces an electric field in the loop of wire driving current. Since the magnetic field is going back and forth, the current goes back and forth, so that is an AC generator.

The fourth of Maxwell's equations is Maxwell's improvement of Ampere's law. This is almost exactly like Faraday's law with the roles of the fields reversed. A time varying electric field will produce a magnetic field. All the previous description about loops and flux through a loop apply. However, once again, just like with the two Gauss' laws, with the roles reversed the equations are not quite the same because there is no magnetic charge. In Ampere's law there is an extra piece. Current, (moving charge), can also induce a magnetic field.

This is how we make electric motors. The current flowing through a loop of wire makes a magnetic field (electro magnet) and that can be used to pull or push on a permanent magnet or another electromagnet.

One implication of the last two laws is light propagation. A time varying electric field creates a time varying magnetic field which in turn creates a time varying electric field etc etc. and the oscillation between the two fields propagates away as light.

I hope that helps.

6. Nov 26, 2016

### FactChecker

7. Nov 29, 2016

### Mary curie

Thanks to all of you guys and again sorry for disturbing you all :)

8. Nov 29, 2016

### Staff: Mentor

In the US, most undergraduates get their first exposure to Maxwell's equations in the second semester of a calculus-based introductory physics course, which often requires only one semester of calculus (basic single-variable derivatives and integrals) as pre-requisite. Textbooks like Halliday/Resnick/Walker (which is what I used as an undergrad, back in the days when it was just Halliday & Resnick) introduce surface, line and volume integrals conceptually and use them to present the integral versions of Maxwell's equations and discuss their significance. They do not attempt to do the differential versions that use divergence and curl.

This doesn't require the detailed calculation techniques that you typically learn in a Calculus III (vector calculus) course. Examples and exercises use very symmetric situations where you can evaluate the integrals by inspecting a diagram and using basic geometrical formulas like the volume and surface area of a sphere or cylinder. I used to call them "Geico integrals: so easy a caveman can do them."

It's a rather limited approach, but it's a start.

9. Nov 29, 2016

### sophiecentaur

I'd normally go all the way with you about 'the Maths' but there are some pretty fair arm waving descriptions of what Maxwell's Equations are about. 'Curl' is the only part of vector analysis at that level that's not very easy without the Maths. The idea of leaves on the surface of a stream, turning round because of the variation of water speed across the surface gives the notion of the Curl operation. It at least can give a student with only elementary Maths knowledge that there's hope for the future.

10. Nov 29, 2016

### Staff: Mentor

But are the students not doing more calculus in parallel with the first and second semesters of that physics course? That's how my undergraduate program worked, and we got to differential and multi-variable vector calculus just in time for Maxwell's equations. (In hindsight, I'm even thinking that the physics professor might have lingered on electrostatics for a bit longer than necessary, just to let the parallel calculus track run a bit longer - the cross-enrollment was near 100%).

11. Nov 29, 2016

### Jonathan Burns

What Maxwell's equations say, on a very basic level, is ...

* Electric charge makes the electric field spread out or close in.

* Moving electric charge (that is, current) makes the magnetic field circulate.

As well as this, the equations say that the two fields act on each other:

* Changing electric field makes the magnetic field circulate.

* Changing magnetic field makes the electric field circulate.

To make this a bit more visual, the two fields have the same form as a
flowing stream -- even though there is nothing flowing, each field has
a direction everywhere, and a certain strength. You can imagine drawing
the streamlines as they follow the path of the river.

There is plain, simple flow where the streamlines just continue on.

There is flow that spreads out, say from a pipe pouring into the river.
This is called divergence. Negative divergence is where the flow
gathers in, say into a pipe going downhill.

There is also flow that circulates in eddies, or changes strength between
the streamlines, as a river slows its speed near the banks. This is called curl.

If we connected two wires to a car battery, and brought the tips close
together, there would be a moderately strong electric field between
them -- say 12 volts. The field would spread out from the tip with
the positive charge, and draw back in to the tip with the negative
charge. If there were positive particles floating in the air, they
would follow the field lines from positive to negative.

This is what the first of Maxwell's equations is saying. The field
diverges from the charge at one tip, and converges to the opposite
charge at the other. In the equation, $\vec E$ is the electric field, the
Del ($\nabla$) sign with a dot stands for divergence, and the q or
$\rho$ sign on the other side is the density of the charge.

Now if the two tips were joined to let charge flow (dangerous!)
then there would also be a magnetic field. Its field lines would
circle around the wires. You could see its direction with a compass
needle.

This is what another of the equations is saying. $\vec B$ is the magnetic
field, the $\nabla \times$ stands for curl, meaning eddying
strength, and the j on the other side is the current density.

Another equation has $\nabla \cdot \vec B$ equal to zero. This means that
there is no divergence or convergence in the magnetic field --
only simple flow or circulation.

There are two more parts to the equations -- these are the ones
with the $\partial$ signs and the t for time. They stand for the
rate at which the fields are changing with time -- as they would
in the instant we joined the wire tips. Change in one field produces
circulation in the other.

Those are the basics, really. The calculus parts tie the equations
into an elegant branch of math, vector calculus, which lets us
calculate the shapes of the streamlines -- either for the electric
and magnetic fields or the flow of a river.

12. Nov 29, 2016

### Staff: Mentor

How the physics and calculus sequences match up, varies from school to school. Here, at the college where I work, we have Calculus I and II as (at least) co-requisites for Physics I and II. I'm pretty sure I remember it being that way at the college where I was an undergraduate, and at the first college where I taught after finishing my Ph.D. At schools like these, it's common for freshmen to take Calculus I and Physics I in fall semester, and Calculus II and Physics II in spring semester.

At the University of Michigan, where I went to grad school, Calculus I and II are pre-requisite for Physics I and II respectively (both the standard and honors versions). There, students would normally take Calculus III (vector calculus) alongside Physics II.

One consideration here is that larger schools offer both Physics I and II every semester, so freshmen can take Physics I in the spring if they have to start with Calculus I in the fall. Small schools usually or often offer Physics I only in the fall and Physics II only in the spring. At those schools, making Calculus I a pre-requisite for Physics I would force students who have to start with Calculus I (i.e. no AP or transfer credit) to wait until the fall of their sophomore year to start with physics.

13. Nov 30, 2016

### Delta²

The science of physics (and pretty much every other science) is like a huge staircase with many turns and twists.

When we view this staircase from a distance it might be terrifying (how am I gonna climb up (i.e. learn) all these stairs???).

However when we come close to study science we see that this staircase consists of many little steps. Learning how to climb each little step will help us climb the staircase. In the staircase of physics many little steps consist of pure mathematical knowledge. The part of staircase that leads to Maxwell's equations is heavily consisting of mathematical steps(integral, differential and vector calculus are the mathematics involved).

One can try to jump over the mathematical steps and try to learn physics without too much math involved. This course of action might work and will provide with a certain degree of qualitative knowledge of physics, but in order to gain a full both qualitative and quantitative knowledge of physics you have to learn the math involved as well.

14. Nov 30, 2016

### Cutter Ketch

Disturbing?? This is how we get our jollies. We get depressed when there are no posts we can answer. Thank you for asking.