An idealized magnetic field that is uniform within a specific...

Consider the rectangular Amperian loop $PQRS$.

The line integral of the magnetic field along the loop is given by:

$\oint \vec{B} \cdot d\vec{l} = \int_P^Q \vec{B} \cdot d\vec{l} + \int_Q^R \vec{B} \cdot d\vec{l} + \int_R^S \vec{B} \cdot d\vec{l} + \int_S^P \vec{B} \cdot d\vec{l}$

Along the paths $PQ$ and $RS$, the magnetic field $\vec{B}$ is perpendicular to the path elements $d\vec{l}$, so $\vec{B} \cdot d\vec{l} = 0$.

Along the path $QR$, the magnetic field is zero, so $\int_Q^R \vec{B} \cdot d\vec{l} = 0$.

Along the path $SP$, the magnetic field $\vec{B}$ is uniform and parallel to $d\vec{l}$. Let the length of $SP$ be $l$. Then $\int_S^P \vec{B} \cdot d\vec{l} = Bl$.

Therefore, the total line integral is $\oint \vec{B} \cdot d\vec{l} = Bl \neq 0$.

However, the current enclosed by the loop $PQRS$ is zero ($I_{enclosed} = 0$).

According to Ampere's law, $\oint \vec{B} \cdot d\vec{l} = \mu_0 I_{enclosed}$. This would require $Bl = 0$, which contradicts the presence of a uniform magnetic field.

Thus, such a field configuration is impossible because the line integral evaluates to a non-zero value, whereas the enclosed current is zero.