Divergence of a Magnetic Field not equaling zero

[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!

 

[haruspex]

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.

 

[Maliska]

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.

 

[haruspex]

A_ρ, 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.

 

[paranormal]

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!