Definition of potential

The electric potential $V$ is the electrostatic potential energy possessed by a test charge, divided by the value of the charge itself. It is thus a scalar quantity, measured in volts ($1\ \text{V} = 1\ \text{J/C}$), which assigns to every point in space a number independent of the charge placed there. Its great advantage is precisely this: while the field $\vec{E}$ is a vector with three components, the potential condenses the same information into a single scalar. Knowing $V$ at every point is equivalent to knowing the field, because $\vec{E}$ can be obtained from the equipotential surfaces and vice versa.

Principle — Electric potential

The potential generated by a point charge $Q$ at distance $r$ is $\ev{V = k\,\frac{Q}{r}}$ The potential difference between two points $A$ and $B$ is the negative of the circulation of the field along a path joining them: $V_B - V_A = -\int_A^B \vec{E}\cdot \dd\vec{\ell}$

The physical intuition is that of a landscape of heights: the potential plays the role of altitude and the charge that of the mass moving through it. A positive charge placed where $V$ is high possesses a great deal of potential energy, exactly like a stone at the top of a hill; left free, it tends to slide towards lower values of $V$, converting potential energy into kinetic energy.

Key formula

$V = k\dfrac{Q}{r}$ $E_\text{pot} = q\,V$ $V_B - V_A = k\,Q\!\left(\dfrac{1}{r_B}-\dfrac{1}{r_A}\right)$

The potential energy of a charge $q$ immersed in the potential $V$ is simply $E_\text{pot} = qV$: the potential is the energy per unit charge, and multiplying it by the actual charge gives the true energy. For the point charge, the potential difference between two distances $r_A$ and $r_B$ is written compactly as $V_B - V_A = kQ\left(1/r_B - 1/r_A\right)$, which shows how $V$ decreases in magnitude with distance from the source and tends to zero at infinity.

Collegamenti

Argomenti: Campo elettrico e potenziale Concetti: Potenziale elettrico · Energia potenziale elettrostatica

Esercizi collegati: Potential difference and work · Drill: work in moving a charge · Two connected conducting spheres