An equipotential surface is perpendicular to:

An equipotential surface is defined as a surface where every point has the same electric potential ($V = \text{constant}$). 1. **Work and Potential Difference**: The work done ($W$) in moving a charge $q$ from one point to another on a surface is given by $W = q(V_2 - V_1)$. 2. **Zero Work**: Since $V_1 = V_2$ for any two points on an equipotential surface, the potential difference $\Delta V = 0$. Therefore, the work done in moving a charge along an equipotential surface is zero ($W = 0$). 3. **Force and Displacement**: Work is also defined by the dot product of the electric force ($\vec{F} = q\vec{E}$) and the displacement vector ($d\vec{s}$): $$W = \int q\vec{E} \cdot d\vec{s} = 0$$ 4. **Perpendicularity**: For the dot product $\vec{E} \cdot d\vec{s}$ to be zero for any displacement $d\vec{s}$ along the surface (where $\vec{E} \neq 0$), the electric field vector $\vec{E}$ must be perpendicular to the displacement vector. Conclusion: Since electric field lines represent the direction of the electric field vector at any point, the electric field lines must be perpendicular to the equipotential surface.