Misplaced Pages

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30-470: The hartree is approximately the negative electric potential energy of the electron in a hydrogen atom in its ground state and, by the virial theorem , approximately twice its ionization energy ; the relationships are not exact because of the finite mass of the nucleus of the hydrogen atom and relativistic corrections . The hartree is usually used as a unit of energy in atomic physics and computational chemistry : for experimental measurements at

60-415: A system containing only one point charge is zero, as there are no other sources of electrostatic force 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 doesn't act to move the point charge itself, it doesn't contribute to

90-423: A system of n charges q 1 , q 2 , …, q n at positions r 1 , r 2 , …, r n respectively, is: U E = 1 2 ∑ i = 1 n q i V ( r i ) . {\displaystyle U_{\mathrm {E} }={\frac {1}{2}}\sum _{i=1}^{n}q_{i}V(\mathbf {r} _{i}).} The electrostatic potential energy of

120-406: Is associated with the configuration of a particular set of point charges within a defined system . An object may be said to have electric potential energy by virtue of either its own electric charge or 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

150-460: Is set to zero when r ref 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

180-769: Is termed as the total work done by an external agent in bringing the charge or the system of charges from infinity to the present configuration without undergoing any acceleration. 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'} } The electrostatic potential energy can also be defined from

210-559: Is the distance between q i and q j . The electrostatic potential energy U E 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 1 generates an electrostatic potential V 1 , which is a function of position r , then U E = q 2 V 1 ( r 2 ) . {\displaystyle U_{\mathrm {E} }=q_{2}V_{1}(\mathbf {r} _{2}).} Doing

240-948: Is the distance between charge Q i and Q j . 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

270-941: Is the distance between the point charges q and Q i , and q and Q i are the assigned values of the charges. The electrostatic potential energy U E stored in a system of N charges q 1 , q 2 , …, q N at positions r 1 , r 2 , …, r N respectively, is: U E = 1 2 ∑ i = 1 N q i V ( r i ) = 1 2 k e ∑ i = 1 N q i ∑ j ≠ i j = 1 N q j r i j , {\displaystyle U_{\mathrm {E} }={\frac {1}{2}}\sum _{i=1}^{N}q_{i}V(\mathbf {r} _{i})={\frac {1}{2}}k_{e}\sum _{i=1}^{N}q_{i}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {q_{j}}{r_{ij}}},} where, for each i value, V( r i )

300-2249: Is the electric potential in r 1 created by charges Q 2 and Q 3 , V ( r 2 ) {\displaystyle V(\mathbf {r} _{2})} is the electric potential in r 2 created by charges Q 1 and Q 3 , and V ( r 3 ) {\displaystyle V(\mathbf {r} _{3})} is the electric potential in r 3 created by charges Q 1 and Q 2 . The potentials are: V ( r 1 ) = V 2 ( r 1 ) + V 3 ( r 1 ) = 1 4 π ε 0 Q 2 r 12 + 1 4 π ε 0 Q 3 r 13 {\displaystyle V(\mathbf {r} _{1})=V_{2}(\mathbf {r} _{1})+V_{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}}}} V ( r 2 ) = V 1 ( r 2 ) + V 3 ( r 2 ) = 1 4 π ε 0 Q 1 r 21 + 1 4 π ε 0 Q 3 r 23 {\displaystyle V(\mathbf {r} _{2})=V_{1}(\mathbf {r} _{2})+V_{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}}}} V ( r 3 ) = V 1 ( r 3 ) + V 2 ( r 3 ) = 1 4 π ε 0 Q 1 r 31 + 1 4 π ε 0 Q 2 r 32 {\displaystyle V(\mathbf {r} _{3})=V_{1}(\mathbf {r} _{3})+V_{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 ij

330-441: Is the electrostatic potential due to all point charges except the one at r i , and is equal to: V ( r i ) = k e ∑ j ≠ i j = 1 N q j r i j , {\displaystyle V(\mathbf {r} _{i})=k_{e}\sum _{\stackrel {j=1}{j\neq i}}^{N}{\frac {q_{j}}{r_{ij}}},} where r ij

SECTION 10

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360-404: Is the separation between the two point charges. The electrostatic potential energy of a system of three charges should not be confused with the electrostatic potential energy of Q 1 due to two charges Q 2 and Q 3 , because the latter doesn't include the electrostatic potential energy of the system of the two charges Q 2 and Q 3 . The electrostatic potential energy stored in

390-443: Is the static dielectric constant. Also, the electron mass is replaced by the effective band mass m ∗ {\displaystyle m^{*}} . The effective hartree in semiconductors becomes small enough to be measured in millielectronvolts (meV). Electric potential energy Electric potential energy is a potential energy (measured in joules ) that results from conservative Coulomb forces and

420-407: 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

450-414: 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}.} One may take the equation for the electrostatic potential energy of

480-416: The atomic scale, the electronvolt (eV) or the reciprocal centimetre (cm) are much more widely used. where: Effective hartree units are used in semiconductor physics where e 2 {\displaystyle e^{2}} is replaced by e 2 / ε {\displaystyle e^{2}/\varepsilon } and ε {\displaystyle \varepsilon }

510-846: The change in electrostatic potential energy, U E , of a point charge q that has moved from the reference position r ref to 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 ref 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: Usually U E

540-499: The charges (not the absolute values of the charges—i.e., 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. 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,

570-1125: 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}}} The electrostatic potential energy, U E , of one point charge q in

600-648: The electric potential as follows: U E ( r ) = q V ( r ) {\displaystyle U_{\mathrm {E} }(\mathbf {r} )=qV(\mathbf {r} )} The SI unit of electric potential energy is joule (named after the English physicist James Prescott Joule ). In the CGS system the erg is the unit of energy, being equal to 10 Joules. Also electronvolts may be used, 1 eV = 1.602×10 Joules. The electrostatic potential energy, U E , of one point charge q at position r in

630-660: 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 d s 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,

SECTION 20

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660-724: 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]} The energy density, or energy per unit volume, d U d V {\textstyle {\frac {dU}{dV}}} , of

690-604: 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 V ( r 1 ) + Q 2 V ( r 2 ) + Q 3 V ( r 3 ) ] {\displaystyle U_{\mathrm {E} }={\frac {1}{2}}\left[Q_{1}V(\mathbf {r} _{1})+Q_{2}V(\mathbf {r} _{2})+Q_{3}V(\mathbf {r} _{3})\right]} Where V ( r 1 ) {\displaystyle V(\mathbf {r} _{1})}

720-455: The presence of n point charges Q i , taking an infinite separation between the charges as the reference position, is: U E ( r ) = q 4 π ε 0 ∑ i = 1 n Q i r i , {\displaystyle U_{E}(r)={\frac {q}{4\pi \varepsilon _{0}}}\sum _{i=1}^{n}{\frac {Q_{i}}{r_{i}}},} where r i

750-444: The presence of a point charge Q , taking an infinite separation between the charges as the reference position, is: U E ( r ) = 1 4 π ε 0 q Q r {\displaystyle U_{E}(\mathbf {r} )={\frac {1}{4\pi \varepsilon _{0}}}{\frac {qQ}{r}}} where r is the distance between the point charges q and Q , and q and Q are

780-442: The same calculation with respect to the other charge, we obtain U E = q 1 V 2 ( r 1 ) . {\displaystyle U_{\mathrm {E} }=q_{1}V_{2}(\mathbf {r} _{1}).} The electrostatic potential energy is mutually shared by q 1 {\displaystyle q_{1}} and q 2 {\displaystyle q_{2}} , so

810-679: The stored energy of the system. Consider bringing a point charge, q , into its final position near a point charge, Q 1 . The electric potential V( r ) due to Q 1 is V ( r ) = k e Q 1 r {\displaystyle V(\mathbf {r} )=k_{e}{\frac {Q_{1}}{r}}} Hence we obtain, the electrostatic potential energy of q in the potential of Q 1 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 1

840-545: 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]} Using

870-425: The term "electrostatic potential energy" is used to describe the potential energy in systems with time-invariant electric fields. The electric potential energy of a system of point charges is defined as the work required to assemble this system of charges by bringing them close together, as in the system from an infinite distance. Alternatively, the electric potential energy of any given charge or system of charges

900-447: The total stored energy is U E = 1 2 [ q 2 V 1 ( r 2 ) + q 1 V 2 ( r 1 ) ] {\displaystyle U_{E}={\frac {1}{2}}\left[q_{2}V_{1}(\mathbf {r} _{2})+q_{1}V_{2}(\mathbf {r} _{1})\right]} This can be generalized to say that the electrostatic potential energy U E stored in

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