When the Curl of a Vector Field is Orthogonal

B = x##\hat{j}## + z##\hat{k}##.In summary, the professor discussed the concept that the curl of a vector field is always perpendicular to itself, using the example of the magnetic vector potential A and the magnetic field B. This is because the dot product of A and the curl of A is equivalent to taking the determinant of a matrix with two identical rows, which results in a determinant of 0. However, this is not always the case, as there are counterexamples where the curl is not orthogonal to itself. The question remains, in what situations is the curl orthogonal to the original vector field, and will these be encountered in an E/M class? It is also worth noting that
  • #1
Harrisonized
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Simple question. It came out of lecture, so it's not homework or anything. My professor said that the curl of a vector field is always perpendicular to itself. The example he gave is that the magnetic vector potential A is always perpendicular to the direction of the magnetic field B. (I haven't seen contrary in Griffifth's so far.) The reason he gave is that if you dot A into the curl of A, you'll end up taking the determinant of matrix with two of the same rows. Therefore, that determinant is 0. Since that is the equivalent of taking the dot product of A and the curl of A, and the curl of A is B, then A and B are orthogonal because their dot product is zero, and only orthogonal vectors give a dot product of 0.

When I learned about the curl back in vector calculus, I was never told any of this. I can't even find in my book where it says that the curl is orthogonal to itself. This is because it's not. A simple google search gave me counterexamples of when the curl is not orthogonal to itself.

However, my question is this: when is the curl orthogonal to the original vector field? Will I ever see such situations in my E/M class?
 
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  • #2
Doesn't sound right to me. You could add arbitrary constants to the components of A without changing the curl, but these constants will change the direction of A.

For a specific example, Let A = y##\hat{i}## + ##\hat{k}##
 

FAQ: When the Curl of a Vector Field is Orthogonal

1. What does it mean for the curl of a vector field to be orthogonal?

When the curl of a vector field is orthogonal, it means that the vector field is perpendicular to the direction of the curl at every point. In other words, the vector field is rotating around a central axis without any outward or inward movement.

2. How is the curl of a vector field calculated?

The curl of a vector field is calculated using the partial derivative operator, denoted by ∇. This operator is applied to each component of the vector field, and the resulting values are used to form a new vector, known as the curl of the vector field.

3. What are some real-life applications of vector fields with orthogonal curl?

Vector fields with orthogonal curl can be found in many real-life applications, such as fluid dynamics, electromagnetics, and weather forecasting. For example, the curl of a wind velocity vector field can indicate the direction and strength of rotation in a tornado or hurricane.

4. Can a vector field have a non-orthogonal curl?

Yes, a vector field can have a non-orthogonal curl. In this case, the vector field is not rotating purely around a central axis, but also has some outward or inward movement. This can occur in situations where there is a net flow or change in the vector field.

5. How does the orthogonality of the curl affect the behavior of a vector field?

The orthogonality of the curl has a significant impact on the behavior of a vector field. It determines the direction and magnitude of rotation at each point in the field, and also affects the overall flow and movement of the vector field. In general, a vector field with an orthogonal curl will exhibit more rotational movement compared to a field with a non-orthogonal curl.

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