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Inductively coupled plasmas (ICP) are well known to exhibit two modes of operation: a low density capacitive E-mode and a high density inductive H-mode. In this study we investigate the E-H transition in a cylindrical ICP, and show the effect of an external magnetic cusp field on the transition dynamics. The plasma is simulated by an electro-magnetic particlein-cell Monte Carlo collision code in order to take into account spatio-temporal variations of the plasma dynamics as well as kinetic effects. Simulations are compared to photometry measurements on the Linac4 H-ion source plasma chamber. We show that the E-H transition is characterized by strong spatial variations of the plasma parameters, with an axial plasma oscillation in E-mode followed by a centring in the coil region in H-mode. The external magnetic cusp field prevents electrons close to the wall to be accelerated and reduces the inductive power deposition in the plasma. This resulted in a approximate to 50% higher current to achieve E-H transition compared to the configuration without cusp field. The results indicate possible improvements to the magnetic cusp field configuration in order to achieve optimal power transfer.