Q9P - Introduction to Electrodynamics

Question

An infinite number of different surfaces can be fit to a given boundary line, and yet, in defining the magnetic flux through a loop, $\varphi =\int B.da$ da, I never specified the particular surface to be used. Justify this apparent oversight.

Step-by-Step Solution

Answer

It is necessary to specify particular surface oversight.

Step 1: Write the given data from the question.

The magnetic flux through the loop, $\varphi =\int B.da$

Step 2: Justify the apparent oversight.

The divergence of the magnetic field,

$\nabla .B=0$

The magnetic field can also be written in the form of the curl of the vector A,

$B=\nabla \times A$

The magnetic flux can be given as the integration of the dot product of vector component of the magnetic field over the vector of component of area.

$\mathit{f}\mathbf{=}\mathbf{\int }\mathit{B}\mathbf{.}\mathit{d}\mathit{a}$

 

Substitute $\nabla \times A$for B  and  $da\hat{n}$ for $\stackrel{\rightharpoonup }{da}$ into above equation.

$$\begin{gathered} \varphi =\int \left(\nabla \times A\right).da\hat{n} \\ \varphi =\int \stackrel{\rightharpoonup }{A}.\stackrel{\rightharpoonup }{dl} \end{gathered}$$

 

From the above discussion it indicates that flux depends on the boundary but on the particular chosen surface.

The  $\nabla .B=0$ means $\varphi \int \stackrel{\rightharpoonup }{B}.\stackrel{\rightharpoonup }{d}a$ is independent on any boundary line. Therefore, it is necessary to specify particular surface oversight.