What do dot and cross product mean in electromagnetics?

[hermtm2]

Hello.

I am taking a fundamentals of electromagnetics.
There are couple of formulas I have been using without understanding the concepts.

$$\nabla$$ $$\cdot$$ B = 0
$$\nabla$$ X E = 0 (curl free)

In those cases, what do dot and cross product mean?


Thanks.

 

[zhermes]

If you've been using those equations in this class, I recommend you get a book on vector calculus to brush up.

The first equation states that the divergence of a magnetic field is zero. "Divergence" is just like it sounds, it's how much a field 'spreads out.' Because there are no magnetic monopoles (individual magnetic charges), magnetic fields are divergence-less ("solenoidal").

The second equation says that the curl of the electric field is zero. Again, 'curl' is an apt name, as it measures the 'circulation' of an electric field. This equation isn't universally true, like the first one. Its significance is essentially that for a static charge distribution, electric fields have to start and end from point charges.

 

[Born2bwire]

They are just vector operators as zhermes described above (and do note that the operator is dependent upon the coordinate system you use). A good book to look at is "Div, Grad, Curl and All That" though any vector calculus book should have a sufficient treatment of the subject.

 

[K^2]

It's really a short hand. These aren't real dot-products or cross-products. However, in flat Cartesian 3-space, the divergence of vector field A is given by:

$$\frac{\partial A_x}{\partial x} + \frac{\partial A_y}{\partial y} + \frac{\partial A_z}{\partial z}$$

That can be thought of as:

$$\left(\frac{\partial}{\partial x}, \frac{\partial}{\partial y}, \frac{\partial}{\partial z}\right) \cdot (A_x, A_y, A_z)$$

Giving shorthand:

$$\nabla \cdot A$$

Similarly, you can write out the $\nabla$ in vector form, use the cross product rules, and end up with Cartesian 3-space expression for curl.

But because you can have divergence and curl in more complicated coordinate systems, and even in non-flat spaces, these really aren't just dot product and cross product. It's a shorthand. And if you have to perform these operations in something like spherical coordinate system, you have to derive a new expression for that, which won't look anything like dot or cross product.

 

[jtbell]

Hello.

I am taking a fundamentals of electromagnetics.

[...]

what do dot and cross product mean?

I have to say this bluntly: somebody screwed up. If you're taking a course in electromagnetism that uses those vector derivatives, then either:

(a) it should require a course in vector calculus as a prerequisite, and you should have taken that course; or

(b) it should teach the basic concepts of vector calculus before proceeding to using them in describing electric and magnetic fields.

 

[hermtm2]

Thanks, K^2.