By means of a two-step unitary transformation, the exact quantum mechanical solution of two-dimensional mesoscopic circuit coupled via inductance is derived. The wavefunction, propagator and expectation value of Hamiltonian are evaluated. The fluctuations of charges and currents are calculated and uncertainty products between charges and their conjugate currents are obtained. We apply our results to the system with special cases of the power source such as the sinusoidal and the sawtooth power source and show that the probability densities oscillate with time due to the power source.