Electric Potential Contents 1 Electric potential 1 1.1 Introduction .............................................. 1 1.2 Electrostatics ............................................. 1 1.2.1 Electric potential due to a point charge ............................ 2 1.3 Generalization to electrodynamics .................................. 2 1.4 Units .................................................. 2 1.5 Galvani potential versus electrochemical potential .......................... 3 1.6 See also ................................................ 3 1.7 References ............................................... 3 2 Voltage 4 2.1 Definition ............................................... 4 2.2 Hydraulic analogy ........................................... 5 2.3 Applications .............................................. 5 2.3.1 Addition of voltages ...................................... 6 2.4 Measuring instruments ........................................ 6 2.5 Typical voltages ............................................ 6 2.6 Galvani potential vs. electrochemical potential ............................ 6 2.7 See also ................................................ 6 2.8 References ............................................... 7 2.9 External links ............................................. 7 3 Electric potential energy 8 3.1 Definition ............................................... 8 3.2 Units .................................................. 8 3.3 Electrostatic potential energy of one point charge ........................... 8 3.3.1 One point charge q in the presence of one point charge Q .................. 8 3.3.2 One point charge q in the presence of n point charges Qi ................... 9 3.4 Electrostatic potential energy stored in a system of point charges ................... 9 3.4.1 Energy stored in a system of one point charge ........................ 9 3.4.2 Energy stored in a system of two point charges ........................ 9 3.4.3 Energy stored in a system of three point charges ....................... 9 3.5 Energy stored in an electrostatic field distribution .......................... 9 i ii CONTENTS 3.6 Energy in electronic elements ..................................... 10 3.7 Notes ................................................. 10 3.8 References .............................................. 10 4 Work (physics) 11 4.1 Units .................................................. 11 4.2 Work and energy ........................................... 11 4.3 Constraint forces ............................................ 12 4.4 Mathematical calculation ....................................... 12 4.4.1 Torque and rotation ...................................... 13 4.5 Work and potential energy ....................................... 13 4.5.1 Path dependence ....................................... 14 4.5.2 Path independence ...................................... 14 4.5.3 Work by gravity ........................................ 14 4.5.4 Work by gravity in space ................................... 14 4.5.5 Work by a spring ....................................... 15 4.5.6 Work by a gas ......................................... 15 4.6 Work–energy principle ........................................ 15 4.6.1 Derivation for a particle moving along a straight line ..................... 16 4.6.2 General derivation of the work–energy theorem for a particle ................ 16 4.6.3 Derivation for a particle in constrained movement ...................... 16 4.6.4 Moving in a straight line (skid to a stop) ........................... 17 4.6.5 Coasting down a mountain road (gravity racing) ....................... 18 4.7 Work of forces acting on a rigid body ................................. 18 4.8 References ............................................... 19 4.9 Bibliography .............................................. 19 4.10 External links ............................................. 19 5 Work (electrical) 20 5.1 Overview ............................................... 20 5.1.1 Qualitative overview ..................................... 20 5.1.2 Mathematical overview .................................... 20 5.2 Electric power ............................................. 21 6 Potential energy 22 6.1 Overview ............................................... 22 6.2 Work and potential energy ....................................... 22 6.2.1 Derivable from a potential .................................. 23 6.2.2 Computing potential energy .................................. 23 6.3 Potential energy for near Earth gravity ................................ 23 6.4 Potential energy for a linear spring .................................. 24 6.5 Potential energy for gravitational forces between two bodies ..................... 24 CONTENTS iii 6.6 Potential energy for electrostatic forces between two bodies ..................... 25 6.7 Reference level ............................................ 25 6.8 Gravitational potential energy ..................................... 25 6.8.1 Local approximation ..................................... 26 6.8.2 General formula ....................................... 26 6.8.3 Why choose a convention where gravitational energy is negative? .............. 27 6.8.4 Uses .............................................. 27 6.9 Chemical potential energy ....................................... 27 6.10 Electric potential energy ........................................ 28 6.10.1 Electrostatic potential energy ................................. 28 6.10.2 Magnetic potential energy ................................... 28 6.11 Nuclear potential energy ........................................ 28 6.12 Forces, potential and potential energy ................................. 29 6.13 Notes ................................................. 29 6.14 References ............................................... 30 6.15 External links ............................................. 30 6.16 Text and image sources, contributors, and licenses .......................... 31 6.16.1 Text .............................................. 31 6.16.2 Images ............................................ 33 6.16.3 Content license ........................................ 34 Chapter 1 Electric potential Not to be confused with Electric potential energy. the field. Two such force fields are the gravitational field and an electric field (in the absence of time-varying mag- An electric potential (also called the electric field poten- netic fields). Such fields must affect objects due to the intrinsic properties of the object (e.g., mass or charge) tial or the electrostatic potential) is the amount of electric potential energy that a unitary point electric charge would and the position of the object. have if located at any point in space, and is equal to the Objects may possess a property known as electric charge work done by an electric field in carrying a unit positive and an electric field exerts a force on charged objects. If charge from infinity to that point. the charged object has a positive charge the force will be According to theoretical electromagnetics, electric poten- in the direction of the electric field vector at that point tial is a scalar quantity denoted by Φ(Phi), ΦE or V, equal while if the charge is negative the force will be in the op- to the electric potential energy of any charged particle at posite direction. The magnitude of the force is given by any location (measured in joules) divided by the charge the quantity of the charge multiplied by the magnitude of of that particle (measured in coulombs). By dividing out the electric field vector. the charge on the particle a remainder is obtained that is a property of the electric field itself. 1.2 Electrostatics This value can be calculated in either a static (time- invariant) or a dynamic (varying with time) electric field at a specific time in units of joules per coulomb (J C−1), Main article: Electrostatics or volts (V). The electric potential at infinity is assumed to be zero. The electric potential at a point r in a static electric field A generalized electric scalar potential is also used E is given by the line integral in electrodynamics when time-varying electromagnetic fields are present, but this can not be so simply calcu- lated. The electric potential and the magnetic vector po- tential together form a four vector, so that the two kinds where C is an arbitrary path connecting the point with of potential are mixed under Lorentz transformations. zero potential to r. When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. In this case, the electric field is conservative and determined by the gradient of the 1.1 Introduction potential: Classical mechanics explores concepts such as force, energy, potential etc. Force and potential energy are di- Then, by Gauss’s law, the potential satisfies Poisson’s rectly related. A net force acting on any object will cause equation: it to accelerate. As an object moves in the direction in which the force accelerates it, its potential energy de- creases: the gravitational potential energy of a cannonball r · E = r · (−∇V ) = −∇2V = ρ/" ; at the top of a hill is greater than at the base of the hill. E E 0 As it rolls downhill its potential energy decreases, being where ρ is the total charge density (including bound translated to motion, inertial (kinetic) energy. charge) and ∇· denotes the divergence. It is possible to define the potential of certain force fields The concept of electric potential is closely linked