A

To find the electromotive force (emf) (\mathcal{E}) produced by the changing magnetic field in the rod AB, we use Faraday's law of electromagnetic induction, which states that the induced emf is equal to the rate of change of magnetic flux through the area swept by the rod.

Step 1: Identify the relevant quantities. The magnetic field strength changes at a rate of \frac{dB}{dt} and has a uniform distribution over the cylindrical area. The length of the rod AB is given as 2l.

Step 2: Calculate the change in magnetic flux (\Phi). The magnetic flux through the rod is given by:

\[ \Phi = B \times A = B \times (2l imes w) \]
where w is the width of the rod, we can neglect because it is constant.

Step 3: The emf can be calculated with Faraday's law as:

\[ \mathcal{E} = - \frac{d\Phi}{dt} = - \frac{d}{dt}(B \times (2l)) \]
\[ \mathcal{E} = -2l \cdot \frac{dB}{dt}
Step 4: Substitute the value of \frac{dB}{dt}. Hence, the emf produced between the ends of the rod is
\[ \mathcal{E} = -2l \cdot \frac{dB}{dt}
Therefore, the correct expression showing the relationship is found and hence, the answer is A.