A point charge is a point particle[1A point particle defines a particle which lacks spatial extension.] which has an electric charge. There are two types of electric charges: positive and negative. The elementary charge, a fundamental physical constant usually denoted as e, is the electric charge carried by a single proton, whereas a single electron has charge −e. In the International System of Units (SI) the elementary charge has a measured value of approximately 1.602 176 6208(98)×10^{−19} C. In the natural system of Hartree atomic units (usually abbreviated “au” or “a.u.”)[2Definition of Hartree atomic units.] the elementary charge functions as the unit of electric charge, that is e is equal to 1.
A point charge or monopole is characterized by a strength
q_{0} and is located at a certain position
r_{0}. The amount of electric charge per unit volume is
given by the charge density, ρ(r), whose SI units are
C⋅m^{−3}. The atomic unit of charge density
(e/a_{0}^{3}, where a_{0} = 0.529 177 210 67(12) × 10^{−10}
m is the radius of the first Bohr orbit for hydrogen) is equal to 1.081 202 3770(67) × 10^{12} C ⋅
m^{−3}. The volume charge density can be expressed by the
Dirac delta function, δ(r −
r_{0}), that is equal to zero everywhere but a single
point r = r_{0}:[4Dirac, P. “The Principles of Quantum Mechanics”
Oxford at the Clarendon Press: 1958.,
5Messiah, A. “Quantum Mechanics”,
Volume I
North-Holland Publishing Company, Amsterdam: 1970,
pp. 179-184 and 468-471.,
6Three-Dimensional Dirac Delta
Function.]
ρ(r) = q_{0} δ(r − r_{0}) | (1.1) |
At the point r = r_{0} the Dirac delta function δ(r − r_{0}) is singular in such a manner that
∫ δ(r − r_{0}) d^{3}r = 1 | (1.2) |
So, integrating eq (1.1) over any volume that contains r_{0} yields the electric charge of the distribution
∫ ρ(r) d^{3}r = q_{0} | (1.3) |
It is assumed that the vector q_{0}r_{0} is pointing towards the charge q_{0} if this is positive, and away from the charge if this is negative.
In the general case of an ensemble of N point charges q_{i} (i = 1, 2, …, N), the total charge Q of the distribution is the sum of the N individual point charges:
Q = N ∑ i=1 q_{i} | (1.4) |
and the total charge density is the sum of the N individual charge densities
ρ(r) = N ∑ i=1 q_{i} δ(r − r_{i}) | (1.5) |
Integrating eq. (1.5) over any volume that contains all the r_{i} yields, because of eq. (1.2), the total charge of the distribution
∫ ρ(r) d^{3}r = N ∑ i=1 q_{i} ∫ δ(r − r_{i}) d^{3}r = Q | (1.6) |
The sum of the vectors q_{i} r_{i} defines the dipole moment vector of the distribution
p = N ∑ i=1 q_{i} r_{i} | (1.7) |
In particular, an ensemble of two point-like particles carrying charges of the same strength q and opposite sign represents an electric dipole. An electric dipole can be characterized by its electric dipole moment, a vector quantity that points from the negative charge (q_{1} = −q at location r_{1} = r_{−}) towards the positive charge (q_{2} = +q at location r_{2} = r_{+}), so that it is defined by the difference
p = q (r_{+} − r_{−}) | (1.8) |
The magnitude of the electric dipole moment is given by the value of the charge q times the distance of the opposite charges. It is a measure of the polarity of the charge distribution, that is of the degree of local charge separation. In other words, a dipole exists when the centers of positive and negative charge distribution do not coincide. Though the position vectors r_{+} and r_{−} are defined with respect to some reference system of coordinates, it is clear from eq. (1.8) that the electric dipole moment of a pair of opposite charges does not depend on the choice of origin. This fact holds also in the general case of eq. (1.7), provided the overall charge of the system is zero. Indeed, if all the position vectors, r_{i}, are moved to a new origin r_{0} (i.e. r_{i} → r_{i} − r_{0}), then eq. (1.7) becomes:
p(r_{0}) = N ∑ i=1 q_{i} (r_{i} − r_{0}) = N ∑ i=1 q_{i} r_{i} − Q r_{0} | (1.9) |
which is independent of r_{0} only if Q = 0.[7Conventional choice of the origin for charged systems.]
The SI units for electric dipole moment are coulomb-meter (C⋅m), however the most commonly used unit is the debye (D), which is a CGS unit. A debye is 10^{−18} esu⋅cm and about 3.336 × 10^{−30} C⋅m. An atomic unit of electric dipole moment is e⋅a_{0} and is about 8.478 × 10^{−29} C⋅m.
A general distribution of electric point-charges may be characterized, in addition to its net charge and its dipole moment, by its electric quadrupole moment and, eventually, by higher order moments. The electric quadrupole moment is a symmetric rank-two tensor (i.e. a 3×3 matrix with 9 components, but because of symmetry only 6 of these are independent).
Θ =
N
∑
i=1
q_{i} r_{i}
r_{i}^{†} =
N
∑
i=1
q_{i}
| (1.10) |
As with any multipole moment, if a lower-order moment (monopole or dipole in this case) is non-zero, then the value of the quadrupole moment depends on the choice of the coordinate origin. If all the position vectors, r_{i} in eq. (1.10), are moved to a new origin r_{0} (i.e. r_{i} → r_{i} − r_{0}), eq. (1.10) becomes:
Θ(r_{0}) =
N
∑
i=1
q_{i} (r_{i} − r_{0})
(r_{i} −
r_{0})^{†} =
N ∑ i=1 q_{i} r_{i} r_{i}^{†} − p r_{0}^{†} − r_{0} p^{†} + Q r_{0} r_{0}^{†} | (1.11) |
which states that Θ is independent of r_{0} (i.e. does not depend on the choice of origin) only if p = 0 and Q = 0. For example, the quadrupole moment of an ensemble of four same-strength point-charges, arranged in a square with alternating signs, is coordinate independent.
The SI units for electric quadrupole moment are C⋅m^{2}, however the most commonly used unit is the buckingham (B), which is a CGS units. A buckingham is 1 × 10^{−26} statC⋅cm^{2} and is equivalent to 1 D⋅Å. An atomic unit of electric quadrupole moment is e⋅a0^{2} and is about 4.486 551 484(28) × 10^{−40} C⋅m^{2}.
“Often it is convenient to ignore the fact that charges come in
packages like electrons and protons, and think of them as being spread out in
a continuous smear − or in a 'distribution,' as it is called. This
is O.K. so long as we are not interested in what is happening on too small a
scale. We describe a charge distribution by the 'charge density,'
ρ(r) ≡ ρ(x,y,z). If the amount
of charge in a small volume ΔV_{1} located at the point
r_{1} is Δq_{1}, then ρ
is defined by”[L1“Coulomb’s law;
superposition”
The Feynman Lectures on Physics, Online
Edition,
Volume II, Chapter 4, Section 4.2]
Δq_{1} = ρ(r_{1}) ΔV_{1} | (2.1) |
As always, the volume integral of the charge density ρ(r) over a region of space V in ℝ^{3} is the charge contained in that region.
q = ∫ V ρ(r) d^{3}r = ∫ V ρ(r) dV = ∭ V ρ(x,y,z) dx dy dz | (2.2) |
In the Euclidean space, the components of the position vector r are the cartesian coordinates x, y and z, and the volume element dV is given by the product of the differentials of the cartesian coordinates, dV = dx dy dz.
Electric charge is the zero-th rank moment of of the distribution. Higher rank electric moments are obtained by evaluation of volume integrals of the type
m^{(n)} = ∫ V ρ(r) r^{(n)} d^{3}r | (2.3) |
The discussion of equation (2.3) is postponed to the next page of the manual.
Electrostatics deals with the circumstance in which nothing depends on the time, i.e. the static case. All charges (and their density) are permanently fixed in space. The basic equation of electrostatics is Gauss' law, which in its differential form
∇ ⋅ E = 4 π ρ(r) | (3.1) |
states that the divergence of the electric field E is proportional to the total electric charge density ρ(r). Eq. (3.1) is written in the natural system of Hartree atomic units. In the SI the right hand side of eq. (3.1) is multiplied by the Coulomb's constant, k_{e}.[2Definition of Hartree atomic units.] From this fundamental equation connecting the charge density with the electric field, the potential Φ of the field can be derived. Introducing the potential Φ in such a way its gradient equals −E,
E = − ∇ Φ | (3.2) |
eq. (3.1) can be rewritten in the form of a Poisson equation[9Poisson's equation.]
∇^{2} Φ = − 4 π ρ(r) | (3.3) |
The nabla symbol ∇ in eqs (3.1-3) indicates a differential vector operator whose components are partial derivative operators. It is most commonly used as a convenient mathematical notation to simplify expressions for the gradient (as in eq. 3.2), divergence (as in eq. 3.1), curl, directional derivative, Laplacian (as in eq. 3.3), Jacobian and tensor derivative.[8The vector differential operator ∇.]
3.1. - The electric potential. A solution of Poisson's equation[9Solution of Poisson's equation.] is given by eq. (3.4)
Φ(r) = ∫ ℝ^{3} ρ(r') |r − r'| d^{3}r' | (3.4) |
which defines the electric potential for a continous charge distribution of density ρ(r). In the case of a discrete charge distribution, where the charges are characterized by a strength q_{i} and are located at certain positions r_{i}, the charge density is given by eq. (1.5). Therefore, inserting eq. (1.5) into eq. (3.4) yields
Φ(r) = N ∑ i=1 q_{i} ∫ ℝ^{3} δ(r' − r_{i}) |r − r'| d^{3}r' = N ∑ i=1 q_{i} |r − r_{i}| = N ∑ i=1 Φ_{i}(r) | (3.5) |
where the second equality holds because of the sifting property of the Dirac delta function.[6The Dirac Delta Function.]
The SI unit of electric potential is the volt (V, in honor of Alessandro Volta), which corresponds to J ⋅ C^{−1} and in terms of SI base units is kg ⋅ m^{2} ⋅ s^{−3} ⋅ A^{−1}. The atomic unit of electric potential (E_{h}/e) has a value of 27.111 386 02(17) V.
3.2. - The electric field. The electric field is a vector quantity generated by a distribution of electric charges, as described by Gauss' law, either in its differential form (3.1) or in its integral form.[10Gauss' flux theorem.] Using the definition of eq. (3.2), an expression of the electric field, E(r), is derived for both a continous charge distribution
E(r) = ∫ ℝ^{3} ρ(r') r − r' |r − r'|^{3} d^{3}r' | (3.6) |
and a discrete charge distribution
E(r) = N ∑ i=1 q_{i} r − r_{i} |r − r_{i}|^{3} = N ∑ i=1 E_{i}(r) | (3.7) |
The SI units of the electric field are newtons per coulomb (N ⋅ C^{−1}) or, equivalently, volts per metre (V ⋅ m^{−1}), which in terms of SI base units are kg ⋅ m ⋅ s^{−3} ⋅ A^{−1}. The atomic unit of electric field (E_{h}/e a_{0}) has a value of 5.142 206 707(32) × 10^{11} V ⋅ m^{−1}.
3.3. - The electric field gradient. The electric field gradient is a tensor quantity generated by a distribution of electric charges. Indeed, a given charge distribution generates an electric potential Φ according to equations (3.4) and (3.5). The gradient of this potential is the negative of the electric field generated, as defined by eq. (3.2). The first derivatives of the field, or the second derivatives of the potential, yields the electric field gradient (EFG) or the hessian of the electric potential.
H(r) = − ∇
E(r)^{†} = ∇
∇^{†} Φ(r) =
| (3.8) |
H_{αβ}(r) = − ∂ E_{β}(r) ∂ r_{α} = ∂^{2} Φ(r) ∂ r_{α} ∂ r_{β} = N ∑ i=1 q_{i} [ 3 (r_{α} − r_{i,α}) (r_{β} − r_{i,β}) |r − r_{i}|^{5} − δ_{αβ} |r − r_{i}|^{3} ] | (3.9) |
Tr H(r) = ∇^{2} Φ(r) = 0 | (3.10) |
The SI units of the electric field gradient are V ⋅ m^{−2}, which in terms of SI base units are kg ⋅ s^{−3} ⋅ A^{−1}. The atomic unit of electric field gradient (E_{h}/e a_{0}^{2}) has a value of 9.717 362 356(60) × 10^{21} V ⋅ m^{−2}.
3.4. - Coulomb's law. The force exerted by one point charge on another acts along the line between the charges. It varies inversely as the square of the distance separating the charges and is proportional to the product of the charges. The force is repulsive if the charges have the same sign and attractive if the charges have opposite signs. If q_{1} is at position r_{1} and q_{2} is at r_{2}, the force F_{12} exerted by q_{1} on q_{2} is the vector
F_{12} = k_{e} q_{1} q_{2} r_{12}^{3} r_{12} | (3.11) |
where r_{12} = r_{2} − r_{1} is the vector pointing from q_{1} to q_{2}, and k_{e} is an experimentally determined positive constant called the Coulomb constant.[2k_{e} in SI units = 8.987 551 787 3681 × 10^{9} kg m^{3} s^{−2} C^{−2}.] The magnitude of vector (3.11), i.e. the magnitude of the electric force exerted by a point charge q_{1} on another point charge q_{2} a distance r_{12} away, is given by
F_{12} = k_{e} q_{1} q_{2} r_{12}^{2} | (3.12) |
In the system of Hartree atomic units the Coulomb constant is unity by definition and therefore it will be omitted hereafter. In accord with Newton’s third law,[11Newton’s third law.] the electrostatic force F_{21} exerted by q_{2} on q_{1} is the negative of F_{12}, i.e. F_{21} = − F_{12}.
It is worth noting that, by combining eqs (3.7) and (3.11), the electrostatic force F_{12} exerted by q_{1} on q_{2} can be written as
F_{12} = q_{2} E_{1}(r_{2}) | (3.13) |
where E_{1}(r_{2}) is the electric field generated by the point charge q_{1} at position r_{2}. According to eq. (3.13), the field E_{1}(r_{2}) can be viewed as the force per unit charge on q_{2} (due to all other charges). In a system of charges, each charge exerts a force given by eq. (3.13) on every other charge. The net force on any charge q is the vector sum of the individual forces exerted on that charge by all the other charges in the system.
F(r) = q E(r) | (3.14) |
where E(r), according to eq. (3.7), is the electric field generated by the system of charges at position r. This follows from the principle of superposition of forces, which states that the force acting on a point charge due to a system of point charges is simply the vector addition of the individual forces acting alone on that point charge due to each one of the charges. The resulting force vector is parallel to the electric field vector at that point, with that point charge removed. Therefore, the force F(r) on a small charge q at position r, due to a system of N discrete charges in vacuum is:
F(r) = q N ∑ 1 q_{i} r − r_{i} |r − r_{i}|^{3} = q N ∑ 1 q_{i} R_{i} |R_{i}|^{3} | (3.15) |
where q_{i} and r_{i} are the magnitude and position respectively of the i-th charge, and the vector R_{i} = r − r_{i} points from charges q_{i} to q. For a continous charge distribution where ρ(r′) gives the charge per unit volume at position r′, the principle of superposition is also used. In this case an integral over the region containing the charge is equivalent to an infinite summation over infinitesimal elements of space containing a point charge dq' = ρ(r′) d^{3}r' (where d V' = d^{3}r' is an infinitesimal element of volume):
F(r) = q ∫ ℝ^{3} ρ(r') r − r' |r − r'|^{3} d^{3}r' | (3.16) |
There are three conditions to be fulfilled for the validity of Coulomb’s
law:[L5Coulomb’s law
From Wikipedia, the free encyclopedia]
The SI unit of force is the newton (N), which in terms of SI base units is kg ⋅ m ⋅ s^{−2}. In atomic units the force is expressed in hartrees per Bohr radius (E_{h}/a_{0}). The atomic unit of force has a value of 8.238 723 36(10) × 10^{−8} N. The CGS unit of force is the dyne, which in in terms of CGS base units is g ⋅ cm ⋅ s^{−2}, so the SI unit of force, the newton, is equal to 10^{5} dynes.
3.5. - Electrostatic potential energy and electrostatic potential energy density. Electrostatic potential energy, U_{E}, is the energy possessed by an electrically charged object because of its position relative to other electrically charged objects.
The electrostatic potential energy, U_{E}, 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 (3.14) to bring q from infinity to position r,
U_{E} = − W_{∞→r} = − q r ∫ ∞ E(r') ⋅ d r' = q r ∫ ∞ ∇ Φ(r') ⋅ d r' = q [Φ(r) − Φ(∞)] = q Φ(r) | (3.17) |
where E is electrostatic field defined by eq. (3.2) and dr' is the displacement vector in a curve from infinity to the final position r. The third equality holds beacause Coulomb forces are conservative, so that the work they do in moving a particle between two points is independent of the taken path. In the limit as r → ∞, the electrostatic potential Φ(r) falls off like 1/r, and is consequently zero. This explains the last equality in eq. (3.17).
U_{E} = ∑ all pairs q_{i} q_{j} r_{ij} = 1 2 N ∑ i=1 q_{i} Φ_{i} | (3.18) |
U_{E} | = 1 2 (U_{E} + U_{E}) |
= 1 2 ( q_{2} q_{1} r_{12} + q_{3} q_{1} r_{13} + q_{3} q_{2} r_{23} + q_{2} q_{1} r_{12} + q_{3} q_{1} r_{13} + q_{3} q_{2} r_{23} ) | |
= 1 2 [ q_{1} ( q_{2} r_{12} + q_{3} r_{13} ) + q_{2} ( q_{3} r_{23} + q_{1} r_{12} ) + q_{3} ( q_{1} r_{13} + q_{2} r_{23} ) ] | |
= 1 2 (q_{1} Φ_{1} + q_{2} Φ_{2} + q_{3} Φ_{3}) |
Equation (3.18) also describes the electrostatic potential energy of a continuous charge distribution. By considering that each volume element dV contains the element of charge dq = ρ dV, eq. (3.18) can be written
U_{E} = 1 2 ∫ all space ρ(r_{1}) ρ(r_{2}) r_{12} dV_{1} dV_{2} = 1 2 ∫ V_{1} ρ(r_{1}) Φ(r_{1}) dV_{1} ≡ 1 2 ∫ V ρ(r) Φ(r) dV | (3.19) |
where the factor ^{1}⁄_{2} is introduced because in the double integral over dV_{1} and dV_{2} all pairs of charge elements have been counted twice. In addition, the second equality in eq. (3.19) follows from the fact that the integral over dV_{2} is just the potential at r_{1} (cf eq. 3.4):
Φ_{1} ≡ Φ(r_{1}) = ∫ V_{2} ρ(r_{2}) r_{12} dV_{2} | (3.20) |
The last identity in eq. (3.19) comes from the fact that the point r_{2} no longer appears in the second equality, so that the suffix 1 is reduntant and can be safely neglected.
Finally, it's worth to note that the electrostatic energy can be described not only in terms of the charges, but also in terms of the electric fields they produce:
U_{E} = ∫ V u_{E} dV = 1 8 π ∫ V E^{2}(r) d^{3}r | (3.21) |
This formula shows that when an electric field E(r) is present, there is located in space an energy U_{E} whose density (energy per unit volume) is proportional to the square of the electric field strength, i.e. u_{E} = E^{2}(r) ⁄ 8π.
U_{E} = − 1 8 π ∫ V Φ ∇^{2}Φ dV | (3.21-1) |
Φ ∇^{2}Φ | = Φ ( ∂^{2} Φ ∂ x^{2} + ∂^{2} Φ ∂ y^{2} + ∂^{2} Φ ∂ z^{2} ) | |
= ∂ ∂ x ( Φ ∂ Φ ∂ x ) − ( ∂ Φ ∂ x )^{2} + ∂ ∂ y ( Φ ∂ Φ ∂ y ) − ( ∂ Φ ∂ y )^{2} + ∂ ∂ z ( Φ ∂ Φ ∂ z ) − ( ∂ Φ ∂ z )^{2} | ||
= ∇ ⋅ (Φ ∇Φ) − (∇Φ) ⋅ (∇Φ) | (3.21-2) |
U_{E} = 1 8 π ∫ V (∇Φ) ⋅ (∇Φ) dV − 1 8 π ∫ V ∇ ⋅ (Φ ∇Φ) dV | (3.21-3) |
∫ V ∇ ⋅ (Φ ∇Φ) dV = ∫ S (Φ ∇Φ) ⋅ n dS | (3.21-4) |
U_{E} = 1 8 π ∫ V (∇Φ) ⋅ (∇Φ) dV = 1 8 π ∫ V E ⋅ E dV | (3.21-5) |
The SI unit of energy is the joule (J), which in terms of SI base units is kg ⋅ m^{2} ⋅ s^{−2}. In atomic units the energy is expressed in hartree (E_{h}). The atomic unit of energy has a value of 4.359 744 17(75) × 10^{−18} J or, in more common non SI units, 27.211 385 eV and 627.509 kcal ⋅ mol^{−1}.
Definition of the Dirac delta function in one dimension,
δ(x − x_{0}), can be found in
reference [5Messiah, A.
“Quantum Mechanics”, Volume I
North-Holland Publishing
Company, Amsterdam: 1970, pp. 179-184 and 468-471.].
Here it is extended to three-dimensional Dirac delta function,
δ_{3}(r −
r_{0}) or simply δ(r
− r_{0}), because of its relevance to
electrostatics. The easiest way to define a three-dimensional delta
function is just to take the product of three one-dimensional functions,
i.e. in cartesian coordinates:
δ(r − r_{0}) = δ(x − x_{0}) δ(y − y_{0}) δ(z − z_{0}) | (N6.1) |
Expressions for cylindrical and spherical coordinates can be be found at
link [L3Weisstein, Eric W.
“Delta Function.”
From MathWorld − A Wolfram Web
Resource.
http://mathworld.wolfram.com/DeltaFunction.html].
The Dirac delta function δ(r −
r_{0}) is defined by the property
∫ f(r) δ(r − r_{0}) d^{3}r = f(r_{0}) | (N6.2) |
for any function f(r) that is continous at the point r = r_{0}. The integral (N6.2) is over any volume that contains the point r = r_{0}. At this point δ(r − r_{0}) is singular in such a manner that
∫
δ(r − r_{0})
d^{3}r = +∞ ∫ −∞ δ(x − x_{0}) dx +∞ ∫ −∞ δ(y − y_{0}) dy +∞ ∫ −∞ δ(z − z_{0}) dz = 1 | (N6.3) |
In addition, δ(r − r_{0}) is zero everywhere except at r = r_{0}, i.e.
δ(r − r_{0}) = 0 ∀ r ≠ r_{0} | (N6.4) |
The discrete analog of the Dirac delta function is the Kronecker delta function, which is usually defined on a discrete domain and takes values 0 and 1.
Eq. (N6.2) is also known as the “sifting” property of the Dirac delta function since that, when the delta function appears in a product within an integrand, it has the property of “sifting” out the value of the integrand at the point where the delta function is not null.
∇ =
∂
∂ r
=
| (N8.1) |
It is a convenient mathematical notation for different differential operators, like the gradient of a function f,
grad f =
∇ f =
| (N8.2) |
the divergence of a vector v,
div v = ∇ ⋅ v = ∇^{†} v = ∂ v_{x} ∂ x + ∂ v_{y} ∂ y + ∂ v_{z} ∂ z | (N8.3) |
the curl of a vector v,
curl v = ∇ ×
v = det
| (N8.4) |
and the divergence of the gradient of a function f, i.e. the laplacian of f,
div grad f = ∇^{†} ∇ f = ∇ ⋅ ∇ f = ∇^{2} f = ∂^{2} f ∂ x^{2} + ∂^{2} f ∂ y^{2} + ∂^{2} f ∂ z^{2} | (N8.5) |
It's worth noting that the gradient operator acts on a scalar differentiable function f = f(r), where r ∈ ℝ^{n} (with n = 3 in eq. N8.2), and returns a vector. Here, the simple notation ∇ is a useful mnemonic device for the product of a vector by a scalar. The divergence operator acts on a vector field v = v(r), where r, v ∈ ℝ^{n} (with n = 3 in eq. N8.3), and returns a scalar function. The notation ∇ ⋅ is a useful mnemonic device for the dot product of two vectors. The curl operator acts on a vector field v = v(r), where r, v ∈ ℝ^{3} (notice that, unlike the gradient and divergence, the curl operator does not generalize in n dimensions), and returns a vector field. The notation ∇ × is a useful mnemonic device for the cross product of two vectors. The Laplacian is a scalar operator that acts on a scalar function and generates a scalar function. The notation ∇^{2} is a shorthand for the dot product of a vector operator with itself, which gives the squared norm of the vector operator.
The Poisson's equation is a partial differential equation (PDE) of the form
∇^{2} u(r) = f(r) | (N9.1) |
which describes the response u(r) of a physical system to an arbitrary source f(r). The general solution of eq. (N9.1) is given by the sum of any harmonic function [which satisfies the homogeneous Laplace equation, ∇^{2} u(r) = 0] and a particular solution of the non-homogeneous Poisson equation (N9.1). Let's first consider the non-homogeneous problem associated with the presence of a point source located at r', mathematically represented by the Dirac delta function δ(r − r'):
∇^{2} G(r, r') = δ(r − r') | (N9.2) |
where the Green's function G(r, r') describes the response of the system at the point r to a point source located at r'.
If G is the fundamental solution of eq. (N9.2), then the convolution
G * f = ∫ ℝ^{3} G(r, r') f(r') d^{3}r' | (N9.3) |
will be a solution of eq. (N9.1). Indeed, because the Laplacian ∇^{2} is a linear differential operator, it follows that
∇^{2}(G * f ) =
(∇^{2} G) * f =
δ * f =
∫ ℝ^{3} δ(r − r') f(r') d^{3}r' = f(r) | (N9.4) |
where the first equality comes from linearity of the Laplacian, the second equality comes from eq. (N9.2), the third equality follows from the definition of convolution, and the last equality is a consequence of the fact that the Dirac delta function is an identity element for convolution, as stated by eq. (N6.2). Therefore, if G is the fundamental solution, the convolution u = G * f is one solution of ∇^{2} u(r) = f(r). This does not mean that it is the only solution. Several solutions for different initial conditions can be found.
The definition (N9.2) of the fundamental solution implies that, if the Laplacian of G is integrated over any volume that encloses the source point, then, according to eq. (N6.3),
∭ V ∇^{2} G dV = 1 | (N9.5) |
The Laplace equation is unchanged under a rotation of coordinates, and hence we can expect that a fundamental solution may be obtained among solutions that only depend upon the distance r from the source point. If we choose the volume to be a ball of radius a around the source point, then Gauss' divergence theorem[10The divergence theorem.] implies that
∭ V ∇^{2} G dV = ∬ S d G d r d S = 4 π a^{2} d G d r |_{r=a} = 1 | (N9.6) |
Rearranging the last equality of eq. (N9.6) yields
d G d r |_{r=a} = 1 4 π a^{2} | (N9.7) |
and hence
G(r, r') = − 1 4 π a = − 1 4 π ⋅ 1 |r − r'| | (N9.8) |
as a^{2} = (x − x')^{2} + (y − y')^{2} + (z − z')^{2} = |r − r'|^{2} is the equation of a sphere of radius a centered at the source point r'. Finally, inserting eq. (N9.8) into eq. (N9.3) gives the solution of Poisson's equation (N9.1):
u(r) = − 1 4 π ∫ ℝ^{3} f(r') |r − r'| d^{3}r' | (N9.9) |
Eq. (N9.9) is a solution of Poisson's equation of electrostatics (3.3), provided that u(r) = Φ(r) and f(r) = − 4 π ρ(r).
The divergence theorem states that the surface integral of the normal component of an arbitrary vector F over a closed surface S is equal to the volume integral of the divergence of the vector over the volume V interior to the surface S:
∬ S (F ⋅ n) dS = ∭ V (∇ ⋅ F) dV | (N10.1) |
where dS may be used as a shorthand for n dS. The left-hand side of the equation represents the total flow across the closed boundary S, and the right-hand side represents the total of the sources in the volume V. In other words, the divergence theorem is a mathematical statement of the physical fact that, in the absence of the creation or destruction of matter, the density within a region of space can change only by having it flow into or away from the region through its boundary. In the special case where the vector F is defined as the gradient of a scalar G, F = ∇ G, and the volume V is that is that of a ball of radius a, the divergence theorem yields eq. (N9.5).
Newton's third law: forces always occur in equal and opposite pairs. If object A exerts a force F_{A,B} on object B, an equal but opposite force F_{B,A} is exerted by object B on object A. Thus,
F_{B,A} = − F_{A,B} | (N11.1) |
This is also known as the Action - Reaction Principle: "For every action there is always an opposite and equal reaction".