  Numerical transfer-matrix methods are applied to two-dimensional Ising spin
systems, in presence of a confining magnetic field which varies with distance
$|{\vec x}|$ to a "trap center", proportionally to $(|{\vec x}|/\ell)^p$,
$p>0$. On a strip geometry, the competition between the "trap size" $\ell$ and
the strip width, $L$, is analysed in the context of a generalized finite-size
scaling {\em ansatz}. In the low-field regime $\ell \gg L$, we use
conformal-invariance concepts in conjunction with a linear-response approach to
derive the appropriate ($p$-dependent) limit of the theory, which agrees very
well with numerical results for magnetization profiles. For high fields $\ell
\lesssim L$, correlation-length scaling data broadly confirms an existing
picture of $p$-dependent characteristic exponents. Standard spin-1/2 and spin-1
Ising systems are considered, as well as the Blume-Capel model.
