## 1 Introduction  
Cement-based materials have been widely used in the construction industry. Thus, many infrastructures consist of concrete. Early detection of internal damage to concrete structure is essential for structural longevity. However, since defects and damage inside concrete structures cannot be seen with the naked eye, it is difficult to detect them. To solve these problems, numerous studies have explored techniques for detecting the conditions inside concrete, such as half-cell and linear polarization resistance measurements, X-ray attenuation measurements, and acoustic emission sensors [1; 2; 3]. However, macro-scale tomography is essential for comprehensively visualizing the condition of concrete structures. Recent studies have explored electrical impedance tomography (EIT), a computerized tomography (CT) technology, for visualizing the internal conditions of concrete structures.  
CT generates a two-dimensional (2D) image of the distribution of specific variables inside an object using data measured at the object's boundary [4]. Medical CT has generally used X-rays as an incident source. However, due to the radiation-shielding characteristics of concrete, X-rays are not an ideal incident source for concrete structures. As another CT technique, EIT utilizes electric sources instead of X-rays. For medical purposes, EIT offersfast imaging with fewer side effect than X-ray-based CT [5]. [PERSON] et al. [6] demonstrated that the advantages of EIT over conventional CT in medical field. As such, EIT was initially developed for medical use.  
Several studies have been conducted to obtain images of cement-based materials using EIT [7; 8; 9]. Because the electrical properties of concrete with cracks, moisture, or ion ingress differ from those of intact concrete, EIT can detect its defects and damage based on changes in these parameters [7; 10; 11]. However, the quality of reconstructions has not been sufficiently high for practical applications. This is because inverse ill-posed problems are extremely sensitive even to relatively small experimental and theoretical errors [7]. Such errors and uncertainties are mainly related to the object's geometry and electrical boundary conditions. Thus, determining the effects of geometry and boundary conditions on the electric potential is key to improving the quality of EIT image reconstructions for concrete applications.  
For medical applications, EIT analytical solutions of the disk (circular section) [12] and rectangular cross section using the Dirac delta function as a current injection model [13] have been investigated. On the other hand, the EIT boundary conditions and analytical solutions for cement-based materials and their representative geometry have not been systematically explored. Instead, the mathematical theories developed for medical use have been applied to cement-based materials. Therefore, fundamental research is necessary to explore analytical solutions for rectangular geometry, which is frequently used for concrete elements, and boundary conditions appropriate for cement-based materials.  
In this study, analytical solutions were mathematically derived for the domain of the rectangular section frequently used in concrete structural elements. A main objective was to determine the appropriate boundary conditions using mathematical models that describe current injection: the Dirac delta function, the Heaviside step function, and the Gaussian function. To validate the analytical solutions and boundary conditions, voltages were measured using EIT and compared to theoretical voltages obtained from various boundary conditions and analytical solutions. A good match between EIT measurements and calculations indicates that the theoretical calculations accurately evaluate the electrical conductivity of cementitious materials. From these comparisons, this study presents appropriate experimental setups for cement-based materials and necessary values for the analytical solutions.