sharepoint enterprise – Adding the choice to the field associated with the content type

I have the following script:

$site = new-object Microsoft.SharePoint.SPSite("http://TEST123")
$web = $site.OpenWeb()
$field = $ct.Fields.GetFieldByInternalName("AddTESTColumn")                            
$ct.FieldLinks($ ="AddTESTColumn"     

I've used the above to update the full name of a column associated with CT.

But now, I want to update a field of choice column to add a new choice "far" to the current choices tried to use the choice.add but continue to get an error message

I used the following:

$site = Get-SPSite -Identity "http://TEST123"

$web = $site.RootWeb
$field = 

and get the following message:

You can not call a method on a NULL value expression.

the problem is with $field = $web.Fields("AddTESTColumn"); as its not recognizing the name of the same field if it exists. I've used both the internal name and the display name, the same problem.