Magnetized discs and photon rings around Yukawa-like black holes

We present stationary solutions of geometrically thick discs (or tori) endowed with a self-consistent toroidal magnetic field distribution surrounding non-rotating black hole in an analytical, static, spherically-symmetric $f(R)$-gravity background. These models introduce Yukawa-like modification to the Newtonian potential, encoded single parameter $\delta$ which controls strength modified potential and whose specific values affect disc configurations when compared general relativistic case. Our span different strengths, from purely hydrodynamical highly magnetized tori. The characteristics are identified by analyzing central density, mass, geometrical size, angular metric deviations Schwarzschild space-time. In limit ($\delta=0$) our reproduce previous results for hole. For small parameter, corresponding $\sim 10\%$ relativity, we find variations 2 \%$ event horizon 5\%$ shift location inner edge center disc, while outer increases 10\%$. analysis $|\delta|>0.1$, however, reveals notable changes space-time solution have major impact morphological thermodynamical properties discs. comparison relativity is further investigated computing size photon ring produced source located at infinity. This allows us place constraints on parameters model based Event Horizon Telescope observations light M87 SgrA$^*$.