Electric Potential Due to a Point Charge
The electric potential of a point charge Q is given by
Learning Objective

Express the electric potential generated by a single point charge in a form of equation
Key Points
 Recall that the electric potential is defined as the potential energy per unit charge, i.e.
$V=\frac{PE}{q}$ .  The potential difference between two points ΔV is often called the voltage and is given by
$\Delta V = V_{B}  V_{A} = \frac{\Delta PE}{q}$ . The potential at an infinite distance is often taken to be zero.  The case of the electric potential generated by a point charge is important because it is a case that is often encountered. A spherical sphere of charge creates an external field just like a point charge, for example.
 The equation for the electric potential due to a point charge is
$V=\frac{kQ}{r}$ , where k is a constant equal to 9.0×10^{9} N⋅m^{2}/C^{2}.
Terms

voltage
The amount of electrostatic potential between two points in space.

electric potential
The potential energy per unit charge at a point in a static electric field; voltage.
Full Text
Electric Potential Due to a Point Charge
Overview
Recall that the electric potential is defined as the electric potential energy per unit charge
The electric potential tells you how much potential energy a single point charge at a given location will have. The electric potential at a point is equal to the electric potential energy (measured in joules) of any charged particle at that location divided by the charge (measured in coulombs) of the particle. Since the charge of the test particle has been divided out, the electric potential is a "property" related only to the electric field itself and not the test particle. Another way of saying this is that because PE is dependent on q, the q in the above equation will cancel out, so V is not dependent on q.
The potential difference between two points ΔV is often called the voltage and is given by
Point Charges
Point charges, such as electrons, are among the fundamental building blocks of matter. Furthermore, spherical charge distributions (like on a metal sphere, see figure below) create external electric fields exactly like a point charge. The electric potential due to a point charge is, thus, a case we need to consider. Using calculus to find the work needed to move a test charge q from a large distance away to a distance of r from a point charge Q, and noting the connection between work and potential (W=–qΔV), it can be shown that the electric potential V of a point charge is
where k is a constant equal to 9.0×10^{9} N⋅m^{2}/C^{2 }.
Van de Graaff Generator
The voltage of this demonstration Van de Graaff generator is measured between the charged sphere and ground. Earth's potential is taken to be zero as a reference. The potential of the charged conducting sphere is the same as that of an equal point charge at its center.
The potential at infinity is chosen to be zero. Thus V for a point charge decreases with distance, whereas E for a point charge decreases with distance squared:
The electric potential is a scalar while the electric field is a vector. Note the symmetry between electric potential and gravitational potential  both drop off as a function of distance to the first power, while both the electric and gravitational fields drop off as a function of distance to the second power.
Key Term Reference
 electric field
 Appears in these related concepts: Maxwell's Predictions and Hertz' Confirmation, Gauss's Law, and Ampere's Law: Magnetic Field Due to a Long Straight Wire
 energy
 Appears in these related concepts: Surface Tension, Energy Transportation, and Introduction to Work and Energy
 equation
 Appears in these related concepts: Equations and Inequalities, Graphs of Equations as Graphs of Solutions, and What is an Equation?
 matter
 Appears in these related concepts: Physical and Chemical Properties of Matter, Introduction: Physics and Matter, and The Study of Chemistry
 potential
 Appears in these related concepts: What is Potential Energy?, Conservative and Nonconservative Forces, and Linear Expansion
 potential difference
 Appears in these related concepts: Energy Conservation, The ElectronVolt, and Principles of Electricity
 potential energy
 Appears in these related concepts: Escape Speed, Defining Graviational Potential Energy, and Types of Energy
 power
 Appears in these related concepts: Force of Muscle Contraction, What is Power?, and Authority
 scalar
 Appears in these related concepts: VectorValued Functions, Superposition of Electric Potential, and Addition and Subtraction; Scalar Multiplication
 symmetry
 Appears in these related concepts: Symmetry of Functions, Balance, and Rhythm
 vector
 Appears in these related concepts: Multiplying Vectors by a Scalar, Series and Sigma Notation, and Translations
 work
 Appears in these related concepts: Heat and Work, Free Energy and Work, and The First Law of Thermodynamics
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