I am having trouble formatting the formula for a quoted vector magnetic potential inside and outside a shell of radius R with charge density sigma

Question

I am having trouble formatting the following formula as a reference to MSU's result for the magnetic field inside a spherical shell and the vector magnetic potential outside the magnetic shell.

The magneti\bigg {\uparrow}c field is given (at Brilliant.org) as:

$$ \overrightarrow{B} = \overrightarrow{\nabla} \times \overrightarrow{A}$$

The vector magnetic potential on the MSU website is (picture version):

That website shows that the magnetic field is constant inside the shell as:

$$\overrightarrow{B} = ({2}/{3}) \mu_0 \sigma \overrightarrow{\omega} $$

I want to format everything within MathJax for Stack Exchange Math instead of using pictures (which is discouraged if at all possible).

I found this reference on the Stack Exchange Website about how to include a large left bracket:

$$ D_{it} = \bigg {if \ bank \ i \ issues \ ABS \ at \ time \ t \\ 2 \ if \ bank \ i \ issue \ CBs \ at \ time \ t \\ \ 0 \ otherwise } $$

But this large left bracket reference is not working for me. Instead of the expected output, I am getting:

Someone helped me to understand the necessary text inside the question to format the ** magnetic potential $\overrightarrow{A}$ **, with the two lines to the right and the text "for points inside the sphere" and "for points outside the sphere" aligned together at the left, to properly reference for the equation.

The example given there is:

$$ f(n) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases} $$

I am pretty sure that this referenced example solves the puzzle, but I am in the middle of testing it. Thank you, the-amplitwist so much for your comment that I just now upvoted.

My attempt to implement it for the referenced formula (in-the-works-still now) is below:

$$ \overrightarrow{A}(\overrightarrow{r}) = \begin{cases} n/2, & \text{if $n$ is even} \\ 3n+1, & \text{if $n$ is odd} \end{cases} $$

My attempt to use *bigg with the new formatting example:

$$ \overrightarrow{A}(\overrightarrow{r}) = \begin{cases} n/2, & \text{if $n$ is even} \\ \\ 3n+1, & \text{if $n$ is odd} \end{cases} $$

$$ \bigg {\uparrow} $$

I think that I am over the major hurdle, but I just want to make sure that the example is complete to reference in the future. I am not sure if the Bigg part is going to work still and I need to test it further. If I need to use something else besides Bigg to get the large left bracket, please let me know. I tried searching for Bigg at the formatting reference, but nothing came up.

I found another reference to bigg for an arrow. The reference works fine as the up-arrow indeed is big. But when I try to apply it to the formula example I still get an error.

If it is possible to help me find out how to get the text larger (so it is easier to read), I would appreciate it.

I am now making another attempt at solving this issue, starting with the approach from a given answer:

$$ \overrightarrow A(\overrightarrow r)= \begin{cases} \frac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere} \\ \frac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere} \end{cases} $$

I am trying to make the inside characters big to force the entire equation to render as big, starting with a previous answer that seems to have been deleted, which was:

I start with the answer from Peter Phipps:

$$ \overrightarrow A(\overrightarrow r)= \begin{cases} \frac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere} \\ \frac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere} \end{cases} $$

Next, I try to make the inside characters big using dfrac instead of frac:

$$ \overrightarrow A(\overrightarrow r)= \begin{cases} \dfrac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere} \\ \dfrac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere} \end{cases} $$

The MathJax code is as follows, starting with a paste and then {} pre-formatted text to get it to quote without the MathJax symbols:

$$
\overrightarrow A(\overrightarrow r)= 
\begin{cases}
\dfrac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere} \\ 
\dfrac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere}
\end{cases}
$$

I am giving Peter Phipps an up-vote because he got me 90% to the final solution. Thank you Peter and community! If the question and answers are helpful, I appreciate your further votes for him too.

Answer 1

This should get you started. As learning anything new it's a matter of experimenting and seeing what you get.

$$
\overrightarrow A(\overrightarrow r)=
\begin{cases}
\frac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere} \\
\frac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere}
\end{cases}
$$

$$ \overrightarrow A(\overrightarrow r)= \begin{cases} \frac{\mu_0 R \sigma}{3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points inside the sphere}\\ \frac{\mu_0 R^4 \sigma}{3r^3} \left(\overrightarrow \omega \times \overrightarrow r\right), \quad\text{for points outside the sphere} \end{cases} $$