Electric potential and voltage Knowledge-Based Study

Study notes

1. To understand electric circuits and electrostatics, it is essential to distinguish between electric potential energy and electric potential.

   Electric potential energy, denoted by $U$, is an extensive property belonging to a system of interacting charges, measured in joules (J). In contrast, electric potential, denoted by $V$, is an intensive property characteristic of a specific location in an electric field, independent of the test charge placed at that location.

   The electric potential $V$ at any point is defined as the electric potential energy per unit positive charge:

   $$V = \frac{U}{q}$$

   Here, $q$ represents the quantity of charge experiencing the potential. The SI unit of electric potential is the volt (V), where:

   $$1\text{ V} = 1\text{ J/C}$$

   A point with an electric potential of 12 V will impart 12 joules of potential energy to every 1 coulomb of positive charge located there.

   In practical applications, we rarely measure absolute potential; instead, we measure the difference in potential between two distinct points, known as electric potential difference or voltage ($\Delta V$).

   When a charge $q$ moves between point A and point B, the change in electric potential is defined as:

   $$\Delta V = V_B - V_A = \frac{\Delta U}{q} = -\frac{W_{\text{field}}}{q}$$

   Key principles governing work and voltage include:

   • Work by the Electric Field: As the electrostatic field performs work $W_{\text{field}}$ on a charge, the system loses potential energy: $\Delta U = -W_{\text{field}}$.
   • Work by an External Agent: To move a charge at constant kinetic energy against the field, an external force must perform work equal to $W_{\text{ext}} = q\Delta V$.
   • Path Independence: Because electrostatic forces are conservative, the work done moving a charge between two points—and therefore $\Delta V$—depends solely on the initial and final positions, not on the path taken.

   The relationship between an electric field $E$ and electric potential $V$ depends on the spatial configuration of the charges.

   1. Uniform Electric Field

   In a uniform electric field, such as between two oppositely charged parallel plates separated by distance $d$, the magnitude of the potential difference is given by:

   $$|\Delta V| = E d \quad \text{or} \quad E = -\frac{\Delta V}{\Delta x}$$

   Electric field lines always point in the direction of decreasing electric potential.

   2. Potential of a Point Charge

   Taking the reference potential to be zero at infinity ($V = 0$ as $r \to \infty$), the electric potential at a distance $r$ from a point charge $Q$ is:

   $$V = \frac{k Q}{r}$$

   Unlike the electric field vector, electric potential is a scalar. For multiple point charges, the total potential is the direct algebraic sum of the individual potentials, taking into account the algebraic signs of each charge.

   3. Equipotential Surfaces

   An equipotential surface consists of all adjacent points having identical electric potential. Because $\Delta V = 0$ along such a surface, no work is required to move a charge across it, and electric field lines are always perpendicular to equipotential surfaces at every point.

   Free charges accelerate naturally under electrostatic forces according to energy conservation principles:

   • Positive charges accelerate spontaneously from regions of higher potential to regions of lower potential, thereby decreasing their electric potential energy and increasing their kinetic energy.
   • Negative charges accelerate spontaneously from regions of lower potential to regions of higher potential, also decreasing their electric potential energy and gaining kinetic energy.

   By applying the conservation of mechanical energy ($\Delta K + \Delta U = 0$), the change in kinetic energy $\Delta K$ is related to the voltage traversed:

   $$\Delta K = -q\Delta V$$

   In atomic and nuclear physics, energy is frequently expressed in electron-volts (eV). One electron-volt is the kinetic energy gained or lost when an elementary charge $e$ traverses a potential difference of 1 volt:

   $$1\text{ eV} = (1.602 \times 10^{-19}\text{ C})(1\text{ V}) = 1.602 \times 10^{-19}\text{ J}$$