Electric potential energy - wikipedia Electric potential energy Not to be confused with Electric potential or Electric power. This article is about the physical magnitude Electric Potential Energy. For electrical energy, see Electrical energy. For energy sources, see Energy development. For electricity generation, see Electricity generation. Electric potential energy Common symbols U SI unit joule ( J ) Derivations from other quantities U = C V / 2 Electromagnetism Electricity Magnetism Electrostatics Electric charge Static electricity Electric field Conductor Insulator Triboelectricity Electrostatic discharge Induction Coulomb 's law Gauss 's law Electric flux / potential energy Electric dipole moment Polarization density Magnetostatics Ampère 's law Magnetic field Magnetization Magnetic flux Biot -- Savart law Magnetic dipole moment Gauss 's law for magnetism Electrodynamics Lorentz force law Electromagnetic induction Faraday 's law Lenz 's law Displacement current Magnetic potential Maxwell 's equations Electromagnetic field Electromagnetic pulse Electromagnetic radiation Maxwell tensor Poynting vector Liénard -- Wiechert potential Jefimenko 's equations Eddy current London equations Mathematical descriptions of the electromagnetic field Electrical network Electric current Electric potential Voltage Resistance Ohm 's law Series circuit Parallel circuit Direct current Alternating current Electromotive force Capacitance Inductance Impedance Resonant cavities Waveguides Covariant formulation Electromagnetic tensor ( stress -- energy tensor ) Four - current Electromagnetic four - potential Scientists Ampère Coulomb Faraday Gauss Heaviside Henry Hertz Lorentz Maxwell Tesla Volta Weber Ørsted Electric potential energy, or electrostatic potential energy, is a potential energy ( measured in joules ) that results from conservative Coulomb forces and is associated with the configuration of a particular set of point charges within a defined system. An object may have electric potential energy by virtue of two key elements : its own electric charge and its relative position to other electrically charged objects. The term `` electric potential energy '' is used to describe the potential energy in systems with time - variant electric fields, while the term `` electrostatic potential energy '' is used to describe the potential energy in systems with time - invariant electric fields. Contents 1 Definition 2 Units 3 Electrostatic potential energy of one point charge 3.1 One point charge q in the presence of one point charge Q 3.2 One point charge q in the presence of n point charges Q 4 Electrostatic potential energy stored in a system of point charges 4.1 Energy stored in a system of one point charge 4.2 Energy stored in a system of two point charges 4.3 Energy stored in a system of three point charges 5 Energy stored in an electrostatic field distribution 6 Energy stored in electronic elements 7 Notes 8 References Definition The electric potential energy of a system of point charges is defined as the work required assembling this system of charges by bringing them close together, as in the system from an infinite distance. The electrostatic potential energy, U, of one point charge q at position r in the presence of an electric field E is defined as the negative of the work W done by the electrostatic force to bring it from the reference position r to that position r. U E ( r ) = − W r r e f → r = − ∫ r r e f r q E ( r ′ ) ⋅ d r ′ ( displaystyle U_ ( mathrm ( E ) ) ( mathbf ( r ) ) = - W_ ( r_ ( \rm ( ref ) ) \rightarrow r ) = - int _ ( ( mathbf ( r ) ) _ ( \rm ( ref ) ) ) ^ ( mathbf ( r ) ) q mathbf ( E ) ( mathbf ( r ' ) ) cdot mathrm ( d ) mathbf ( r ' ) ), where E is the electrostatic field and dr ' is the displacement vector in a curve from the reference position r to the final position r. The electrostatic potential energy can also be defined from the electric potential as follows : The electrostatic potential energy, U, of one point charge q at position r in the presence of an electric potential Φ ( displaystyle scriptstyle Phi ) is defined as the product of the charge and the electric potential. U E ( r ) = q Φ ( r ) ( displaystyle U_ ( mathrm ( E ) ) ( mathbf ( r ) ) = q Phi ( mathbf ( r ) ) ), where Φ ( displaystyle scriptstyle Phi ) is the electric potential generated by the charges, which is a function of position r. Units The SI unit of electric potential energy is the joule ( named after the English physicist James Prescott Joule ). In the CGS system the erg is the unit of energy, being equal to 10 J. Also electronvolts may be used, 1 eV = 1.602 × 10 J. Electrostatic potential energy of one point charge One point charge Q in the presence of one point charge Q A point charge q in the electric field of another charge Q. The electrostatic potential energy, U, of one point charge q at position r in the presence of a point charge Q, taking an infinite separation between the charges as the reference position, is : U E ( r ) = k e q Q r ( displaystyle U_ ( E ) ( r ) = k_ ( e ) ( \frac ( qQ ) ( r ) ) ), where k e = 1 4 π ε 0 ( displaystyle k_ ( e ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ) is Coulomb 's constant, r is the distance between the point charges q & Q, and q & Q are the charges ( not the absolute values of the charges. For example, an electron would have a negative value of charge when placed in the formula ). The following outline of proof states the derivation from the definition of electric potential energy and Coulomb 's law to this formula. Outline of proof The electrostatic force F acting on a charge q can be written in terms of the electric field E as F = q E ( displaystyle mathbf ( F ) = q mathbf ( E ) ), By definition, the electrostatic potential energy, U, of one point charge q at position r in the presence of an electric field E is the negative of the work done by the electrostatic force to bring it from the reference position r to that position r. U E ( r ) − U E ( r r e f ) = − W r r e f → r = − ∫ r r e f r q E ⋅ d s ( displaystyle U_ ( E ) ( r ) - U_ ( E ) ( r_ ( \rm ( ref ) ) ) = - W_ ( r_ ( \rm ( ref ) ) \rightarrow r ) = - int _ ( ( r ) _ ( \rm ( ref ) ) ) ^ ( r ) q mathbf ( E ) cdot mathrm ( d ) mathbf ( s ) ). where : r = position in 3d space of the charge q, using cartesian coordinates r = ( x, y, z ), taking the position of the Q charge at r = ( 0, 0, 0 ), the scalar r = r is the norm of the position vector, ds = differential displacement vector along a path C going from r to r, W r r e f → r ( displaystyle scriptstyle W_ ( r_ ( \rm ( ref ) ) \rightarrow r ) ) is the work done by the electrostatic force to bring the charge from the reference position r to r, Usually U is set to zero when r is infinity : U E ( r r e f = ∞ ) = 0 ( displaystyle U_ ( E ) ( r_ ( \rm ( ref ) ) = infty ) = 0 ) so U E ( r ) = − ∫ ∞ r q E ⋅ d s ( displaystyle U_ ( E ) ( r ) = - int _ ( infty ) ^ ( r ) q mathbf ( E ) cdot mathrm ( d ) mathbf ( s ) ) When the curl ∇ × E is zero, the line integral above does not depend on the specific path C chosen but only on its endpoints. This happens in time - invariant electric fields. When talking about electrostatic potential energy, time - invariant electric fields are always assumed so, in this case, the electric field is conservative and Coulomb 's law can be used. Using Coulomb 's law, it is known that the electrostatic force F and the electric field E created by a discrete point charge Q are radially directed from Q. By the definition of the position vector r and the displacement vector s, it follows that r and s are also radially directed from Q. So, E and ds must be parallel : E ⋅ d s = E ⋅ d s cos ⁡ ( 0 ) = E d s ( displaystyle mathbf ( E ) cdot mathrm ( d ) mathbf ( s ) = mathbf ( E ) cdot mathrm ( d ) mathbf ( s ) cos ( 0 ) = E mathrm ( d ) s ) Using Coulomb 's law, the electric field is given by E = E = 1 4 π ε 0 Q s 2 ( displaystyle mathbf ( E ) = E = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q ) ( s ^ ( 2 ) ) ) ) and the integral can be easily evaluated : U E ( r ) = − ∫ ∞ r q E ⋅ d s = − ∫ ∞ r 1 4 π ε 0 q Q s 2 d s = 1 4 π ε 0 q Q r = k e q Q r ( displaystyle U_ ( E ) ( r ) = - int _ ( infty ) ^ ( r ) q mathbf ( E ) cdot mathrm ( d ) mathbf ( s ) = - int _ ( infty ) ^ ( r ) ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( qQ ) ( s ^ ( 2 ) ) ) ( \rm ( d ) ) s = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( qQ ) ( r ) ) = k_ ( e ) ( \frac ( qQ ) ( r ) ) ) One point charge Q in the presence of N point charges Q Electrostatic potential energy of q due to Q and Q charge system : U E = q 1 4 π ε 0 ( Q 1 r 1 + Q 2 r 2 ) ( displaystyle U_ ( E ) = q ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) left ( ( \frac ( Q_ ( 1 ) ) ( r_ ( 1 ) ) ) + ( \frac ( Q_ ( 2 ) ) ( r_ ( 2 ) ) ) \right ) ) The electrostatic potential energy, U, of one point charge q in the presence of n point charges Q, taking an infinite separation between the charges as the reference position, is : U E ( r ) = k e q ∑ i = 1 n Q i r i ( displaystyle U_ ( E ) ( r ) = k_ ( e ) q sum _ ( i = 1 ) ^ ( n ) ( \frac ( Q_ ( i ) ) ( r_ ( i ) ) ) ), where k e = 1 4 π ε 0 ( displaystyle k_ ( e ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ) is Coulomb 's constant, r is the distance between the point charges q & Q, and q & Q are the signed values of the charges. Electrostatic potential energy stored in a system of point charges The electrostatic potential energy U stored in a system of N charges q, q,..., q at positions r, r,..., r respectively, is : U E = 1 2 ∑ i = 1 N q i Φ ( r i ) = 1 2 ∑ i = 1 N q i ∑ j = 1 N ( j ≠ i ) k e q j r i j ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 2 ) ) sum _ ( i = 1 ) ^ ( N ) q_ ( i ) Phi ( mathbf ( r ) _ ( i ) ) = ( \frac ( 1 ) ( 2 ) ) sum _ ( i = 1 ) ^ ( N ) q_ ( i ) sum _ ( j = 1 ) ^ ( N ( j \neq i ) ) k_ ( e ) ( \frac ( q_ ( j ) ) ( r_ ( ij ) ) ) ), ( 1 ) where, for each i value, Φ ( r ) is the electrostatic potential due to all point charges except the one at r, and is equal to : Φ ( r i ) = ∑ j = 1 N ( j ≠ i ) k e q j r i j ( displaystyle Phi ( mathbf ( r ) _ ( i ) ) = sum _ ( j = 1 ) ^ ( N ( j \neq i ) ) k_ ( e ) ( \frac ( q_ ( j ) ) ( mathbf ( r ) _ ( ij ) ) ) ), where r is the distance between q and q. Outline of proof The electrostatic potential energy U stored in a system of two charges is equal to the electrostatic potential energy of a charge in the electrostatic potential generated by the other. That is to say, if charge q generates an electrostatic potential Φ, which is a function of position r, then U E = q 2 Φ 1 ( r 2 ). ( displaystyle U_ ( mathrm ( E ) ) = q_ ( 2 ) Phi _ ( 1 ) ( mathbf ( r ) _ ( 2 ) ). ) Doing the same calculation with respect to the other charge, we obtain U E = q 1 Φ 2 ( r 1 ). ( displaystyle U_ ( mathrm ( E ) ) = q_ ( 1 ) Phi _ ( 2 ) ( mathbf ( r ) _ ( 1 ) ). ) The electrostatic potential energy is mutually shared by q 1 ( displaystyle q_ ( 1 ) ) and q 2 ( displaystyle q_ ( 2 ) ), so the total stored energy is U E = 1 2 ( q 2 Φ 1 ( r 2 ) + q 1 Φ 2 ( r 1 ) ) ( displaystyle U_ ( E ) = ( \frac ( 1 ) ( 2 ) ) left ( q_ ( 2 ) Phi _ ( 1 ) ( mathbf ( r ) _ ( 2 ) ) + q_ ( 1 ) Phi _ ( 2 ) ( mathbf ( r ) _ ( 1 ) ) \right ) ) This can be generalized to say that the electrostatic potential energy U stored in a system of N charges q, q,..., q at positions r, r,..., r respectively, is : U E = 1 2 ∑ i = 1 N q i Φ ( r i ) ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 2 ) ) sum _ ( i = 1 ) ^ ( N ) q_ ( i ) Phi ( mathbf ( r ) _ ( i ) ) ). Energy stored in a system of one point charge The electrostatic potential energy of a system containing only one point charge is zero, as there are no other sources of electrostatic potential against which an external agent must do work in moving the point charge from infinity to its final location. A common question arises concerning the interaction of a point charge with its own electrostatic potential. Since this interaction does n't act to move the point charge itself, it does n't contribute to the stored energy of the system. Energy stored in a system of two point charges Consider bringing a point charge, q, into its final position in the vicinity of a point charge, Q. The electrostatic potential Φ ( r ) due to Q is Φ ( r ) = k e Q 1 r ( displaystyle Phi ( r ) = k_ ( e ) ( \frac ( Q_ ( 1 ) ) ( r ) ) ) Hence we obtain, the electric potential energy of q in the potential of Q as U E = 1 4 π ε 0 q Q 1 r 1 ( displaystyle U_ ( E ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( qQ_ ( 1 ) ) ( r_ ( 1 ) ) ) ) where r is the separation between the two point charges. Energy stored in a system of three point charges The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q due to two charges Q and Q, because the latter does n't include the electrostatic potential energy of the system of the two charges Q and Q. The electrostatic potential energy stored in the system of three charges is : U E = 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 3 r 23 ) ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) left ( ( \frac ( Q_ ( 1 ) Q_ ( 2 ) ) ( r_ ( 12 ) ) ) + ( \frac ( Q_ ( 1 ) Q_ ( 3 ) ) ( r_ ( 13 ) ) ) + ( \frac ( Q_ ( 2 ) Q_ ( 3 ) ) ( r_ ( 23 ) ) ) \right ) ) Outline of proof Using the formula given in ( 1 ), the electrostatic potential energy of the system of the three charges will then be : U E = 1 2 ( Q 1 Φ ( r 1 ) + Q 2 Φ ( r 2 ) + Q 3 Φ ( r 3 ) ) ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 2 ) ) left ( Q_ ( 1 ) Phi ( mathbf ( r ) _ ( 1 ) ) + Q_ ( 2 ) Phi ( mathbf ( r ) _ ( 2 ) ) + Q_ ( 3 ) Phi ( mathbf ( r ) _ ( 3 ) ) \right ) ) Where Φ ( r 1 ) ( displaystyle Phi ( mathbf ( r ) _ ( 1 ) ) ) is the electric potential in r created by charges Q and Q, Φ ( r 2 ) ( displaystyle Phi ( mathbf ( r ) _ ( 2 ) ) ) is the electric potential in r created by charges Q and Q, and Φ ( r 3 ) ( displaystyle Phi ( mathbf ( r ) _ ( 3 ) ) ) is the electric potential in r created by charges Q and Q. The potentials are : Φ ( r 1 ) = Φ 2 ( r 1 ) + Φ 3 ( r 1 ) = 1 4 π ε 0 Q 2 r 12 + 1 4 π ε 0 Q 3 r 13 ( displaystyle Phi ( mathbf ( r ) _ ( 1 ) ) = Phi _ ( 2 ) ( mathbf ( r ) _ ( 1 ) ) + Phi _ ( 3 ) ( mathbf ( r ) _ ( 1 ) ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 2 ) ) ( r_ ( 12 ) ) ) + ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 3 ) ) ( r_ ( 13 ) ) ) ) Φ ( r 2 ) = Φ 1 ( r 2 ) + Φ 3 ( r 2 ) = 1 4 π ε 0 Q 1 r 21 + 1 4 π ε 0 Q 3 r 23 ( displaystyle Phi ( mathbf ( r ) _ ( 2 ) ) = Phi _ ( 1 ) ( mathbf ( r ) _ ( 2 ) ) + Phi _ ( 3 ) ( mathbf ( r ) _ ( 2 ) ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 1 ) ) ( r_ ( 21 ) ) ) + ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 3 ) ) ( r_ ( 23 ) ) ) ) Φ ( r 3 ) = Φ 1 ( r 3 ) + Φ 2 ( r 3 ) = 1 4 π ε 0 Q 1 r 31 + 1 4 π ε 0 Q 2 r 32 ( displaystyle Phi ( mathbf ( r ) _ ( 3 ) ) = Phi _ ( 1 ) ( mathbf ( r ) _ ( 3 ) ) + Phi _ ( 2 ) ( mathbf ( r ) _ ( 3 ) ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 1 ) ) ( r_ ( 31 ) ) ) + ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) ( \frac ( Q_ ( 2 ) ) ( r_ ( 32 ) ) ) ) Where r is the distance between charge Q and Q. If we add everything : U E = 1 2 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 1 r 21 + Q 2 Q 3 r 23 + Q 3 Q 1 r 31 + Q 3 Q 2 r 32 ) ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 2 ) ) ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) left ( ( \frac ( Q_ ( 1 ) Q_ ( 2 ) ) ( r_ ( 12 ) ) ) + ( \frac ( Q_ ( 1 ) Q_ ( 3 ) ) ( r_ ( 13 ) ) ) + ( \frac ( Q_ ( 2 ) Q_ ( 1 ) ) ( r_ ( 21 ) ) ) + ( \frac ( Q_ ( 2 ) Q_ ( 3 ) ) ( r_ ( 23 ) ) ) + ( \frac ( Q_ ( 3 ) Q_ ( 1 ) ) ( r_ ( 31 ) ) ) + ( \frac ( Q_ ( 3 ) Q_ ( 2 ) ) ( r_ ( 32 ) ) ) \right ) ) Finally, we get that the electrostatic potential energy stored in the system of three charges : U E = 1 4 π ε 0 ( Q 1 Q 2 r 12 + Q 1 Q 3 r 13 + Q 2 Q 3 r 23 ) ( displaystyle U_ ( mathrm ( E ) ) = ( \frac ( 1 ) ( 4 pi \varepsilon _ ( 0 ) ) ) left ( ( \frac ( Q_ ( 1 ) Q_ ( 2 ) ) ( r_ ( 12 ) ) ) + ( \frac ( Q_ ( 1 ) Q_ ( 3 ) ) ( r_ ( 13 ) ) ) + ( \frac ( Q_ ( 2 ) Q_ ( 3 ) ) ( r_ ( 23 ) ) ) \right ) ) Energy stored in an electrostatic field distribution The energy density, or energy per unit volume, d U d V ( displaystyle ( \frac ( dU ) ( dV ) ) ), of the electrostatic field of a continuous charge distribution is : u e = d U d V = 1 2 ε 0 E 2. ( displaystyle u_ ( e ) = ( \frac ( dU ) ( dV ) ) = ( \frac ( 1 ) ( 2 ) ) \varepsilon _ ( 0 ) left ( mathbf ( E ) ) \right ^ ( 2 ). ) Outline of proof One may take the equation for the electrostatic potential energy of a continuous charge distribution and put it in terms of the electrostatic field. Since Gauss 's law for electrostatic field in differential form states ∇ ⋅ E = ρ ε 0 ( displaystyle mathbf ( \nabla ) cdot mathbf ( E ) = ( \frac ( \rho ) ( \varepsilon _ ( 0 ) ) ) ) where E ( displaystyle mathbf ( E ) ) is the electric field vector ρ ( displaystyle \rho ) is the total charge density including dipole charges bound in a material ε 0 ( displaystyle \varepsilon _ ( 0 ) ) is the permittivity of free space, then, U = 1 2 ∫ all space ρ ( r ) Φ ( r ) d V = 1 2 ∫ all space ε 0 ( ∇ ⋅ E ) Φ d V ( displaystyle ( \begin ( aligned ) U& = ( \frac ( 1 ) ( 2 ) ) int limits _ ( \text ( all space ) ) \rho ( r ) Phi ( r ) , dV & = ( \frac ( 1 ) ( 2 ) ) int limits _ ( \text ( all space ) ) \varepsilon _ ( 0 ) ( mathbf ( \nabla ) cdot ( mathbf ( E ) ) ) Phi , dV end ( aligned ) ) ) so, now using the following divergence vector identity ∇ ⋅ ( A B ) = ( ∇ ⋅ A ) B + A ⋅ ( ∇ B ) ⇒ ( ∇ ⋅ A ) B = ∇ ⋅ ( A B ) − A ⋅ ( ∇ B ) ( displaystyle \nabla cdot ( ( mathbf ( A ) ) ( B ) ) = ( \nabla cdot ( mathbf ( A ) ) ) ( B ) + ( mathbf ( A ) ) cdot ( \nabla ( B ) ) Rightarrow ( \nabla cdot ( mathbf ( A ) ) ) ( B ) = \nabla cdot ( ( mathbf ( A ) ) ( B ) ) - ( mathbf ( A ) ) cdot ( \nabla ( B ) ) ) we have U = ε 0 2 ∫ all space ∇ ⋅ ( E Φ ) d V − ε 0 2 ∫ all space ( ∇ Φ ) ⋅ E d V ( displaystyle U = ( \frac ( \varepsilon _ ( 0 ) ) ( 2 ) ) int limits _ ( \text ( all space ) ) mathbf ( \nabla ) cdot ( mathbf ( E ) Phi ) dV - ( \frac ( \varepsilon _ ( 0 ) ) ( 2 ) ) int limits _ ( \text ( all space ) ) ( mathbf ( \nabla ) Phi ) cdot mathbf ( E ) dV ) using the divergence theorem and taking the area to be at infinity where Φ ( ∞ ) = 0 ( displaystyle Phi ( infty ) = 0 ) U = ε 0 2 ∫ of space boundary Φ E ⋅ d A ⏞ 0 − ε 0 2 ∫ all space ( − E ) ⋅ E d V = ∫ all space 1 2 ε 0 E 2 d V. ( displaystyle ( \begin ( aligned ) U& = overbrace ( ( \frac ( \varepsilon _ ( 0 ) ) ( 2 ) ) int limits _ ( ( ) _ ( \text ( of space ) ) ^ ( \text ( boundary ) ) ) Phi mathbf ( E ) cdot d mathbf ( A ) ) ^ ( 0 ) - ( \frac ( \varepsilon _ ( 0 ) ) ( 2 ) ) int limits _ ( \text ( all space ) ) ( - mathbf ( E ) ) cdot mathbf ( E ) , dV & = int limits _ ( \text ( all space ) ) ( \frac ( 1 ) ( 2 ) ) \varepsilon _ ( 0 ) left ( mathbf ( E ) ) \right ^ ( 2 ) , dV. end ( aligned ) ) ) So, the energy density, or energy per unit volume d U d V ( displaystyle ( \frac ( dU ) ( dV ) ) ) of the electrostatic field is : u e = 1 2 ε 0 E 2. ( displaystyle u_ ( e ) = ( \frac ( 1 ) ( 2 ) ) \varepsilon _ ( 0 ) left ( mathbf ( E ) ) \right ^ ( 2 ). ) Energy stored in electronic elements The electric potential energy stored in a capacitor is U = 1⁄2 CV Some elements in a circuit can convert energy from one form to another. For example, a resistor converts electrical energy to heat. This is known as the Joule effect. A capacitor stores it in its electric field. The total electric potential energy stored in a capacitor is given by U E = 1 2 Q V = 1 2 C V 2 = Q 2 2 C ( displaystyle U_ ( E ) = ( \frac ( 1 ) ( 2 ) ) QV = ( \frac ( 1 ) ( 2 ) ) CV ^ ( 2 ) = ( \frac ( Q ^ ( 2 ) ) ( 2C ) ) ) where C is the capacitance, V is the electric potential difference, and Q the charge stored in the capacitor. Outline of proof One may assemble charges to a capacitor in infinitesimal increments, d q → 0 ( displaystyle dq \to 0 ), such that the amount of work done to assemble each increment to its final location may be expressed as W q = V d q = q C d q. ( displaystyle W_ ( q ) = Vdq = ( \frac ( q ) ( C ) ) dq. ) The total work done to fully charge the capacitor in this way is then W = ∫ d W = ∫ 0 Q V d q = 1 C ∫ 0 Q q d q = Q 2 2 C. ( displaystyle W = int dW = int _ ( 0 ) ^ ( Q ) Vdq = ( \frac ( 1 ) ( C ) ) int _ ( 0 ) ^ ( Q ) qdq = ( \frac ( Q ^ ( 2 ) ) ( 2C ) ). ) where Q ( displaystyle Q ) is the total charge on the capacitor. This work is stored as electrostatic potential energy, hence, W = U E = Q 2 2 C. ( displaystyle W = U_ ( E ) = ( \frac ( Q ^ ( 2 ) ) ( 2C ) ). ) Notably, this expression is only valid if d q → 0 ( displaystyle dq \to 0 ), which holds for many - charge systems such as large capacitors having metallic electrodes. For few - charge systems the discrete nature of charge is important. The total energy stored in a few - charge capacitor is U E = Q 2 C ( displaystyle U_ ( E ) = ( \frac ( Q ^ ( 2 ) ) ( C ) ) ) which is obtained by a method of charge assembly utilizing the smallest physical charge increment Δ q = e ( displaystyle Delta q = e ) where e ( displaystyle e ) is the elementary unit of charge and Q = N e ( displaystyle Q = Ne ) where N ( displaystyle N ) is the total number of charges in the capacitor. The total electrostatic potential energy may also be expressed in terms of the electric field in the form U E = 1 2 ∫ V E ⋅ D d V ( displaystyle U_ ( E ) = ( \frac ( 1 ) ( 2 ) ) int _ ( V ) mathrm ( E ) cdot mathrm ( D ) dV ) where D ( displaystyle mathrm ( D ) ) is the displacement of the electric field within a dielectric material and integration is over the entire volume of the dielectric. The total electrostatic potential energy stored within a charged dielectric may also be expressed in terms of a continuous volume charge, ρ ( displaystyle \rho ), U E = 1 2 ∫ V ρ Φ d V ( displaystyle U_ ( E ) = ( \frac ( 1 ) ( 2 ) ) int _ ( V ) \rho Phi dV ) where integration is over the entire volume of the dielectric. These latter two expressions are valid only for cases when the smallest increment of charge is zero ( d q → 0 ( displaystyle dq \to 0 ) ) such as dielectrics in the presence of metallic electrodes or dielectrics containing many charges. Notes Jump up ^ The reference zero is usually taken to be a state in which the individual point charges are very well separated ( `` are at infinite separation '' ) and are at rest. Jump up ^ Alternatively, it can also be defined as the work W done by an external force to bring it from the reference position r to some position r. Nonetheless, both definitions yield the same results. Jump up ^ The factor of one half accounts for the ' double counting ' of charge pairs. For example, consider the case of just two charges. References Jump up ^ Electromagnetism ( 2nd edition ), I.S. Grant, W.R. Phillips, Manchester Physics Series, 2008 ISBN 0 - 471 - 92712 - 0 Jump up ^ Halliday, David ; Resnick, Robert ; Walker, Jearl ( 1997 ). `` Electric Potential ''. Fundamentals of Physics ( 5th ed. ). John Wiley & Sons. ISBN 0 - 471 - 10559 - 7. 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Question: the si unit for electric potential energy is equivalent to
Answer: joule (J)