Final answer:
To obtain a vector field potential function by inspection, one analyzes the properties and symmetry of a system to infer the potential without integration, seizing on the scalar nature of electric potential (V) and its relationship to the vector electric field (E).
Step-by-step explanation:
Inspecting a vector field to obtain a potential function involves understanding the relationship between the electric potential (V) and the electric field (E). The electric potential is a scalar quantity and can be integrated to calculate the potential energy in a system, while the electric field is a vector quantity that can be derived by taking the gradient of the electric potential. To find a vector field potential function by inspection, one essentially investigates the symmetry and properties of a system to directly infer the potential function without the need for explicit integration.
When considering systems with charged particles, superposition allows us to sum the individual potentials from point charges to find the total potential. On the other hand, calculating the total electric field requires vector addition, taking into account both magnitude and direction. Essential Knowledge 2.A.1 states that a vector field gives the value of a physical quantity that is described by a vector, which in this case relates to the electric field and how it is derived from the scalar potential.