# What do dot and cross product mean in electromagnetics?

• hermtm2
In summary, the conversation discusses the use of two equations in electromagnetics - \nabla \cdot B = 0 and \nabla X E = 0 - which involve vector operators such as divergence and curl. The purpose of these operators is to measure the spread and circulation of electric and magnetic fields. However, if the individual is unfamiliar with vector calculus, it is recommended to brush up on the subject before using these equations.
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.

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.

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.

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.

hermtm2 said:
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.

Thanks, K^2.

## 1. What is the dot product in electromagnetics?

The dot product in electromagnetics is a mathematical operation that combines two vectors to produce a scalar quantity. In other words, it gives a single numerical value that represents the magnitude of the component of one vector in the direction of the other vector.

## 2. How is the dot product used in electromagnetics?

The dot product is used in electromagnetics to calculate the work done by a force on a charged particle moving in an electric or magnetic field. It is also used to determine the angle between two vectors, which is important in understanding the direction of electromagnetic forces.

## 3. What does the cross product represent in electromagnetics?

The cross product in electromagnetics is a mathematical operation that combines two vectors to produce a third vector. This new vector is perpendicular to both of the original vectors and represents the direction of the resulting electromagnetic force.

## 4. How is the cross product used in electromagnetics?

The cross product is used in electromagnetics to calculate the force exerted on a charged particle moving through a magnetic field. It is also used to determine the torque exerted on a current-carrying wire in a magnetic field, which is important in understanding the behavior of electric motors and generators.

## 5. What is the difference between the dot and cross product in electromagnetics?

The main difference between the dot and cross product in electromagnetics is that the dot product results in a scalar quantity, while the cross product results in a vector quantity. Additionally, the dot product is used to determine the component of one vector in the direction of another, while the cross product is used to determine the perpendicular force between two vectors.

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