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Magnetic resonance current density imaging (MRCDI) is to provide current density images of a subject using a magnetic resonance imaging (MRI) scanner with a current injection apparatus. The injection current generates a magnetic field that we can measure from MR phase images. We obtain internal current density images from the measured magnetic flux densities via Ampere's law. However, we must rotate the subject to acquire all of the three components of the induced magnetic flux density. This subject rotation is impractical in clinical MRI scanners when the subject is a human body. In this paper, we propose a way to eliminate the requirement of subject rotation by careful mathematical analysis of the MRCDI problem. In our new MRCDI technique, we need to measure only one component of the induced magnetic flux density and reconstruct both cross-sectional conductivity and current density images without any subject rotation.