1. **Problem Statement:** We have a set of parallel equipotential surfaces with voltages decreasing from 90 V to 40 V from left to right. An electron moves along five different paths between these surfaces. We need to find: (a) The direction of the electric field. (b) Whether the work done on the electron along each path is positive, negative, or zero. (c) Rank the paths according to the work done, greatest first. 2. **Relevant Concepts and Formulas:** - The electric field $\vec{E}$ points from regions of higher potential to lower potential. - Work done by an external force moving a charge $q$ from potential $V_i$ to $V_f$ is $W = q(V_f - V_i)$. - For an electron, charge $q = -e$ (negative). - Positive work means energy is supplied to the electron; negative work means energy is extracted. 3. **Step-by-step Analysis:** **(a) Direction of the electric field:** - Since potential decreases from left (90 V) to right (40 V), the electric field points from left to right. **(b) Work done on the electron for each path:** - Electron charge $q = -e$. - Work done $W = q(V_f - V_i) = -e(V_f - V_i)$. Calculate $V_f - V_i$ for each path: - Path 1: from 70 V to 40 V, $V_f - V_i = 40 - 70 = -30$ $W_1 = -e(-30) = +30e$ (positive work) - Path 2: from 70 V to 80 V, $V_f - V_i = 80 - 70 = +10$ $W_2 = -e(10) = -10e$ (negative work) - Path 3: from 90 V to 80 V, $V_f - V_i = 80 - 90 = -10$ $W_3 = -e(-10) = +10e$ (positive work) - Path 4: from 60 V to 50 V, $V_f - V_i = 50 - 60 = -10$ $W_4 = -e(-10) = +10e$ (positive work) - Path 5: from 80 V to 60 V, $V_f - V_i = 60 - 80 = -20$ $W_5 = -e(-20) = +20e$ (positive work) **(c) Ranking paths by work done (greatest first):** - $W_1 = +30e$ - $W_5 = +20e$ - $W_3 = +10e$ - $W_4 = +10e$ - $W_2 = -10e$ Ranking: Path 1 > Path 5 > Path 3 = Path 4 > Path 2 4. **Summary:** - Electric field points from left to right (higher to lower potential). - Work done is positive when electron moves to lower potential, negative when moving to higher potential. - Path 1 requires the greatest work done on the electron, Path 2 the least (negative).