lec 4

Electric Potential

Definition of Electric Potential

The electric potential at a point is defined as the electric potential energy (EPE) per unit charge at that point. It is a scalar quantity and is measured in volts (V), where 1 V = 1 J/C.

• Formula: $V = \frac{EPE}{q_0}$

• SI Unit: Joule per Coulomb (J/C) = Volt (V)

• Only differences in electric potential (ΔV) are physically meaningful, similar to potential energy.

Analogy: The definition of electric potential is analogous to the definition of the electric field, $E = \frac{F}{q_0}$.

Electric Potential Difference

The electric potential difference (ΔV) between two points is the work done per unit charge by the electric field in moving a test charge between those points.

• Formula: $\Delta V = V_B - V_A = \frac{\Delta EPE}{q_0} = -\frac{W_{BA}}{q_0}$

• Potential difference is often referred to as "voltage".

Energy Conservation: Electric potential energy is included in the total energy of a system, along with kinetic and other forms of potential energy.

• Total Energy: $E = KE_{trans} + KE_{rot} + PE_{grav} + PE_{elastic} + PE_{electric}$

Electron Volt (eV)

The electron volt (eV) is a convenient unit of energy in atomic and nuclear physics. It is the energy gained by an electron moving through a potential difference of one volt.

• Conversion: $1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J}$

• Common multiples: keV (103 eV), MeV (106 eV), GeV (109 eV)

Calculating Electric Potential Difference

Work Done by Electric Field

For a uniform electric field, the work done in moving a charge q0 a distance d is:

• $W_{AB} = Fd = q_0Ed$

• For a parallel-plate capacitor: $E = \frac{Q}{A\varepsilon_0}$

• Potential difference: $\Delta V = -Ed$

Example: Parallel-Plate Capacitor

Calculate the potential difference between the plates of a parallel-plate capacitor with charge $Q = 4.0\ \text{nC}$, area $A = 0.01\ \text{m}^2$, and separation $d = 1.0\ \text{mm}$:

• $\Delta V = -\frac{Qd}{A\varepsilon_0} = -\frac{4.0 \times 10^{-9} \times 0.001}{0.01 \times 8.85 \times 10^{-12}} = -45\ \text{V}$

Electric Potential Due to Point Charges

Potential Created by a Point Charge

The electric potential at a distance r from a point charge q is given by:

• $V = k\frac{q}{r}$, where $k = 8.99 \times 10^9\ \text{N} \cdot \text{m}^2/\text{C}^2$

Example: Potential of a Point Charge

Find the potential at a point 1.2 m from a charge of $4.0 \times 10^{-8}$ C:

• For a positive charge: $V = k\frac{q}{r} = 8.99 \times 10^9 \frac{4.0 \times 10^{-8}}{1.2} = 300\ \text{V}$

• For a negative charge: $V = -300\ \text{V}$

Superposition Principle for Electric Potential

When multiple point charges are present, the total electric potential at a point is the algebraic sum of the potentials due to each charge:

• $V_{total} = \sum_i k\frac{q_i}{r_i}$

Special Case: Electric Dipole

For two equal and opposite charges separated by a distance d, the potential at the midpoint between them is zero:

• $V = k\frac{q}{r} + k\frac{-q}{r} = 0$

Conceptual Example: Where is the Potential Zero?

For two charges, +2q and –q, fixed in place, there are two points along the line joining them where the total potential is zero. This is found by solving:

• $V = k\frac{2q}{r_1} - k\frac{q}{r_2} = 0$

Equipotential Surfaces and Their Relation to the Electric Field

Definition and Properties

An equipotential surface is a surface on which the electric potential is constant everywhere. The electric field is always perpendicular to equipotential surfaces and points in the direction of decreasing potential.

• No work is done by the electric field when moving a charge along an equipotential surface.

• In equilibrium, conductors are equipotential surfaces.

Equipotential Surfaces for Parallel Plates

For a parallel-plate capacitor, equipotential surfaces are planes parallel to the plates. The electric field is uniform and perpendicular to these surfaces.

Relationship Between Electric Field and Potential Gradient

The electric field is related to the rate of change of electric potential with distance (potential gradient):

• $E = -\frac{\Delta V}{\Delta s}$

For a parallel-plate capacitor, this simplifies to:

• $E = \frac{V}{d}$

Example: Spacing Between Equipotential Surfaces

Given a potential difference of –3.0 V between two equipotential surfaces and an electric field of $2.0 \times 10^3$ V/m, the spacing is:

• $\Delta s = -\frac{\Delta V}{E} = -\frac{-3.0}{2.0 \times 10^3} = 1.5 \times 10^{-3}$ m

Summary Table: Key Equations and Concepts

Concept	Equation	Notes
Electric Potential	$V = \frac{EPE}{q_0}$	Scalar quantity, units: V (J/C)
Potential Difference	$\Delta V = -\frac{W_{BA}}{q_0}$	Work done per unit charge
Point Charge Potential	$V = k\frac{q}{r}$	Reference: $V=0$ at infinity
Superposition	$V_{total} = \sum_i k\frac{q_i}{r_i}$	Add potentials algebraically
Electron Volt	$1\ \text{eV} = 1.60 \times 10^{-19}\ \text{J}$	Energy unit (not SI)
Potential Gradient	$E = -\frac{\Delta V}{\Delta s}$	Direction: decreasing V
Parallel Plate Capacitor	$E = \frac{V}{d}$	Uniform field between plates