How Is Magnetic Flux Calculated Through a Disk in a Dipole Field?

[Monocles]

Homework Statement

Consider a thin disk normal to the z-axis, of thickness dz and radius p centered on the axis of a dipole.

a. Show that

$$0 = \oint \vec{B} \cdot d\vec{A} \approx (2\pi p dz)B_{r} + (\pi p^{2}) a (dB_{z} / dz)$$, so $$B_{p} \approx (-p/2)(dB_{z} / dz)$$

b. With B_z for a dipole, find B_p near the axis (small p).

Homework Equations

$$0 = \oint \vec{B} \cdot d\vec{A}$$

The Attempt at a Solution

I don't know where to start because the book doesn't bother to define B_r, B_z, B_p, or a. Is there some standard definition for these? I thought it might be partial derivatives, but this is an intro physics class and we have yet to use partial derivatives anywhere and the book went over the math needed to do things.

 

[AvroArrow]

Could someone explain what B_p is and what a is? I'm pretty sure I know how to do the integral, but I can't do it without knowing what these terms mean.

 

[thomate1]

As a scientist, it is important to always define our variables and equations clearly. In this case, B_r, B_z, B_p, and a are not defined in the given context. However, based on my understanding of the problem, it seems that B_r refers to the radial component of the magnetic field, B_z refers to the axial component of the magnetic field, B_p refers to the magnetic field perpendicular to the disk, and a is a constant related to the dipole moment.

In order to solve this problem, we can use the definition of magnetic flux, which is the integral of the magnetic field over a surface. In this case, the surface is the disk and the magnetic field is the dipole field. We can also use the fact that the magnetic field of a dipole is given by B = (μ0/4π)(3(m⃗ ⋅r̂)r̂ − m⃗), where μ0 is the permeability of free space, m⃗ is the dipole moment, and r̂ is the unit vector in the direction of the position vector r.

a. Using these equations, we can write the magnetic flux through the disk as:

Φ = ∫∫ B⃗ ⋅ dA⃗ = ∫∫ (μ0/4π)(3(m⃗ ⋅r̂)r̂ − m⃗)⋅dA⃗

Since the disk is normal to the z-axis, we can choose a coordinate system such that z is along the axis of the dipole and r is in the radial direction. This means that r̂ = (x̂, ŷ, 0) and dA⃗ = (dx dy, 0, 0). Substituting these values, we get:

Φ = ∫∫ (μ0/4π)(3mz(xdx + ydy) − mx̂dx − mŷdy)

Since the disk is thin, we can approximate dx and dy as dz and p dθ respectively, where θ is the angle from the z-axis. This gives us:

Φ ≈ (μ0/4π) ∫∫ (3mzpz dθ dz − mx̂p dz − mŷp dθ)

Using