# Visuo-Haptic Simulations to Understand the Dependence of Electric Forces on Distance
## 3 Methodology
### Design of Electric Force Visuo-Haptic Scenarios
#### 3.2.2. _Visuo-Haptic Scenarios_  
Three visuo-haptic scenarios were designed: (i) point charge, (ii) line charge, and (iii) plane charge. 1. **Point charge**. The goal of this simulator is that the student identifies the dependence of the electric force between two-point charges as a function of their magnitude, sign, and distance. In Figure 3, a screenshot of the simulator is shown. The user will feel the force strength between a charge \(Q_{1}\) fixed at the coordinate origin and another charge \(Q_{2}\) that can be manipulated with the haptic device. The signs and magnitudes of both charges can be changed. Positive charges are shown in red color while negative ones are shown in blue. The magnitudes of the electrical charges (in \(\mu\)C) can be selected with a sliding bar or entering them directly. The electric force that \(Q_{1}\) exerts on \(Q_{2}\) is represented by an arrow. The user can explore the changes of the feedback force on \(Q_{2}\) by approaching or moving \(Q_{2}\) away from \(Q_{1}\). The screen displays the coordinates of \(Q_{2}\), its distance to \(Q_{1}\), and the components of the electric force. It is worth noting that the length of the displayed force arrow corresponds to the actual magnitude of the force which is calibrated in this case under a \(1/r^{2}\) law. The electric force on \(Q_{2}\) exerted by \(Q_{1}\) is given by the Equation (1). \[\vec{F}=\frac{Q_{1}Q_{2}}{4\pi\varepsilon_{0}r^{2}}\hat{r}\] (1) where \(\varepsilon_{0}\) is the electric permittivity of free space, \(r\) is the distance between the charges, and \(\hat{r}\) is the unit vector connecting the two charges. **Line charge**. The goal of this simulator is that the student recognizes the dependence of the electric force exerted by a very long, straight-line charge distribution on a point charge, as a function of the linear charge density of the line \(\lambda\), the magnitude of the point charge \(Q\), their signs, and the distance between them. Figure 4 presents a screenshot of this visuo-haptic simulator. The line charge is fixed along the vertical axis. The user can manipulate the position of the point charge \(Q\) with the haptic device, and the signs and values of \(\lambda\) and \(Q\) can also be varied. As before, positive charges are shown in red color while negative ones are shown in blue. The magnitudes of the point electric charge (in \(\upmu\)C) and of the linear change (in \(\upmu\)C/m) can be selected with a sliding bar or entering them directly. An arrow drawn on \(Q\) shows the magnitude and direction of the force acting on it. The screen also displays the coordinates of charge \(Q\), its radial distance to the line, and the components and magnitude of the electric force on \(Q\). The length of the displayed arrow for this scenario is proportional to its actual magnitude as returned by the haptic device, according to a programmed \(1/r\) inverse law. The electric force on \(Q\) exerted by the line charge is given by Equation (2). \[\vec{F}=\frac{\lambda Q}{2\pi\varepsilon_{0}r}\hat{r}\] (2) where \(\varepsilon_{0}\) is the electric permittivity of free space, \(r\) is the radial distance between the line charge and the point charge, and \(\hat{r}\) is the corresponding unit vector in the radial direction. **Plane charge**. The goal of this simulator is that the student tests the dependence of the electric force exerted by an infinite charged plane on a point charge as a function of the plane surface charge density \(\sigma\), the charge magnitude \(Q\), their signs, and the distance between them. Figure 5 shows a setup of the haptic device and the visuo-haptic simulator. The flat infinite plane charge is fixed to the \(YZ\) plane. The user can manipulate the point charge \(Q\) with the haptic device and feel the force acting on it. The signs and values of \(\sigma\) and \(Q\) can be varied and an arrow shows the magnitude and direction of the electric force exerted on \(Q\). Positive charges are shown in red while negative ones are shown in blue.