Get Workspace By Capacity Name

Capacity names are meaningless.  Depending on your tenant admin rights you can either see all workspaces, or you can see the workspaces that you are a member of.

Groups - Get Groups (Power BI Power BI REST APIs) | Microsoft Docs

# Calls the Active Directory Authentication Library (ADAL) to authenticate against AAD
function GetAuthToken
{
    if(-not (Get-Module AzureRm.Profile)) {
        Import-Module AzureRm.Profile
    }
    $clientId = "your client id here"
    $redirectUri = "urn:ietf:wg:oauth:2.0:oob"
    $resourceAppIdURI = "https://analysis.windows.net/powerbi/api"
    $authority = "https://login.microsoftonline.com/common/oauth2/authorize";
    $authContext = New-Object "Microsoft.IdentityModel.Clients.ActiveDirectory.AuthenticationContext" -ArgumentList $authority
    $authResult = $authContext.AcquireToken($resourceAppIdURI, $clientId, $redirectUri, "Auto")
    return $authResult
}

# Get the auth token from AAD
$token = GetAuthToken

# Building Rest API header with authorization token
$authHeader = @{
    'Content-Type'='application/json'
    'Authorization'=$token.CreateAuthorizationHeader()
}

# get workspaces with capacityId where present

$uri = "https://api.powerbi.com/v1.0/myorg/groups"
$v = (Invoke-RestMethod -Uri $uri –Headers $authHeader –Method GET).value 
$v | Select id,isReadOnly,isOnDedicatedCapacity,capacityId,name | Format-Table -AutoSize