Divergence of a Magnetic Field not equaling zero

• Maliska
In summary, the conversation discusses the discrepancy between the proof of ∇°B = 0 in cylindrical coordinates and the definition of divergence in those coordinates. The speaker is seeking clarification on how a vector with constant components in Cartesian coordinates can have nonzero partial derivatives in cylindrical coordinates, specifically the component in the ρ direction. The other person in the conversation suggests that the unit ρ-direction vector may be the source of the discrepancy.
Maliska
I have a larger problem involving divergences and curls, but the correct answer requires ∇°B (divergence of B) = 0. I understand the proof of this in Griffiths, but the definition of divergence in cylindrical coordinates is:

After using the product rule to split the first term, we get the divergence of B is B_rho / rho + 0 + 0 + 0, or simply Div(B)=B_rho / rho; however, this clearly contradicts what we know about the divergence of B being zero. Can someone please clarify this for me, I've been stuck at it for hours.

Thanks.

P.s. First post!

How do you know that the partial derivatives you claim to be zero are zero? Do you have a specific field in mind? A field that's a constant vector in Cartesian can look rather different in polar.

Thank you haruspex for responding. The partial derivatives I said equal zero could be my mistake, but please explain how a vector with constant components in cartesian coordinates can have nonzero partial derivatives in cylindrical coordinates.

Aρ, for example, refers to the component of the vector in the ρ direction. If the vector is constant, its component in the ρ direction may yet depend on θ and z. It's not that the vector varies, but that the unit ρ-direction vector does.

As a fellow scientist, I can understand your frustration with this problem. The concept of divergence and its application in different coordinate systems can be quite tricky. Let me try to clarify this for you.

Firstly, it is important to note that the divergence of a vector field represents the net flow of the field out of a given point. In other words, it tells us how much the field is spreading out or converging at a particular point. Therefore, a non-zero divergence at a point would indicate that there is a net flow of the field out of that point.

Now, in cylindrical coordinates, the divergence of a vector field B can be written as Div(B) = (1/r)(∂(rB_r)/∂r) + (1/r)(∂B_θ/∂θ) + (∂B_z/∂z). This can be derived using the product rule, as you have correctly shown in your post.

However, in the case of a magnetic field, we know that B_θ = 0 and B_z = 0, as the magnetic field lines are always tangent to the surface of a cylinder. Therefore, the only non-zero term in the above equation would be (∂(rB_r)/∂r). Now, if we substitute this in the equation, we get Div(B) = (1/r)(∂(rB_r)/∂r). And since B_r is a function of r, this term would also be zero.

Hence, we can conclude that in cylindrical coordinates, the divergence of a magnetic field is indeed equal to zero, as expected. I hope this explanation helps to clarify your doubts. Keep up the good work and happy researching!

What is the concept of divergence in a magnetic field?

Divergence in a magnetic field refers to the measurement of the amount of magnetic flux flowing out or into a specific point in space. It tells us about the strength and direction of the magnetic field at that point.

Why would the divergence of a magnetic field not be equal to zero?

The divergence of a magnetic field would not equal zero if there is a source or sink of magnetic flux present. This could be due to a magnetic charge or current, or the presence of a magnetic material such as a ferromagnet.

How is the divergence of a magnetic field calculated?

The divergence of a magnetic field is calculated using the vector calculus operator known as the divergence operator. It involves taking the dot product of the magnetic field vector with the del operator, resulting in a scalar value.

What are the implications of a non-zero divergence in a magnetic field?

A non-zero divergence in a magnetic field indicates the presence of a magnetic source or sink. This can have various implications depending on the specific situation, such as affecting the behavior of charged particles or influencing the dynamics of an electromechanical system.

Can the divergence of a magnetic field be negative?

Yes, the divergence of a magnetic field can be negative. This indicates that the magnetic flux is flowing into a point in space, rather than out of it. It is also possible for the divergence to be zero, which would mean that the magnetic field is uniform and does not have a source or sink.

Replies
5
Views
1K
Replies
2
Views
1K
Replies
2
Views
420
Replies
3
Views
629
Replies
1
Views
1K
Replies
4
Views
2K
Replies
2
Views
12K
Replies
1
Views
1K
Replies
12
Views
1K
Replies
1
Views
1K