The Poisson equation , named after the French mathematician and physicist Siméon Denis Poisson , is an elliptical partial differential equation of the second order that is used as part of boundary value problems in large parts of physics.
Mathematical formulation
The Poisson equation is general
![- \ Delta u = f](https://wikimedia.org/api/rest_v1/media/math/render/svg/3c1d2f433a6dc29a5230a45ed7609be30ea24cbb)
Here designated
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the Laplace operator
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the solution you are looking for
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a function. If the equation becomes Laplace's equation .![f \ equiv 0](https://wikimedia.org/api/rest_v1/media/math/render/svg/f3f8c50ad49a75e73abba74c35712a860f5bfa47)
To solve Poisson's equation, additional information must be given, e.g. B. in the form of a Dirichlet boundary condition :
![{\ displaystyle {\ begin {cases} - \ Delta u & = f & {\ text {in}} & \ Omega \\\ quad u & = g & {\ text {on}} & \ partial \ Omega \ end {cases}} }](https://wikimedia.org/api/rest_v1/media/math/render/svg/baf0f296f107d3426e57e9c79285058b5ade0f03)
with open and restricted.
![\ Omega \ subset \ mathbb {R} ^ {n}](https://wikimedia.org/api/rest_v1/media/math/render/svg/79774d994aac0be34ef390915fed12cbce816f6f)
In this case, we construct a solution using the fundamental solution of Laplace's equation:
![\ Phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
![{\ displaystyle \ Phi (x): = {\ begin {cases} - {\ dfrac {1} {2 \ pi}} \ ln | x | & n = 2 \\ {\ dfrac {1} {(n-2 ) \ omega _ {n}}} \ cdot {\ dfrac {1} {| x | ^ {n-2}}} & n \ geq 3 \ end {cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/f49cb3987ba211e3296cc51df95a64d336a38512)
It denotes the area of the unit sphere in the -dimensional Euclidean space .
![{\ displaystyle \ omega _ {n} = {\ tfrac {2 \ pi ^ {\ frac {n} {2}}} {\ Gamma ({\ frac {n} {2}})}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5412a79bc1b3332e2f6e9f7c86af85b4dedbc837)
![n](https://wikimedia.org/api/rest_v1/media/math/render/svg/a601995d55609f2d9f5e233e36fbe9ea26011b3b)
The convolution gives a solution to Poisson's equation.
![(\ Phi * f)](https://wikimedia.org/api/rest_v1/media/math/render/svg/7445addbd6b07985f5595727b0c141af37e6f41b)
In order to meet the boundary condition can be the Green's function use
![G (x, y): = \ Phi (yx) - \ phi ^ {x} (y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/d4e6aafa3be7fac4a13e047eb7557a0c45f9eacf)
is a correction function that
![{\ displaystyle {\ begin {cases} \ Delta \ phi ^ {x} = 0 & {\ text {in}} \ \ Omega \\\ phi ^ {x} = \ Phi (yx) & {\ text {on} } \ \ partial \ Omega \ end {cases}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0019c9c405bea458e894cecc507d5aa8d1fac6b1)
Fulfills. It is generally dependent on and is easy to find only for simple areas.
![\Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/24b0d5ca6f381068d756f6337c08e0af9d1eeb6f)
If one knows , then a solution of the boundary value problem from above is given by
![G (x, y)](https://wikimedia.org/api/rest_v1/media/math/render/svg/3197a4895281863ac09cb8c7c4371d21e93d11f4)
![{\ displaystyle u (x) = - \ int _ {\ partial \ Omega} g (y) {\ frac {\ partial G (x, y)} {\ partial n}} \ mathrm {d} \ sigma (y ) + \ int _ {\ Omega} f (y) G (x, y) \ mathrm {d} y}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a763596f2f170017caccdac93570e123a584984d)
where the surface dimension denotes.
![\ sigma](https://wikimedia.org/api/rest_v1/media/math/render/svg/59f59b7c3e6fdb1d0365a494b81fb9a696138c36)
![\ partial \ Omega](https://wikimedia.org/api/rest_v1/media/math/render/svg/16feddaad462c2a1c9efdaeee062a0484a023fde)
The solution can also be found with the help of the platform method or a variation approach .
Applications in physics
The Poisson equation, for example, is satisfied by the electrostatic potential and the gravitational potential , each with symbols . The function is proportional to the electrical charge density or to the mass density![{\ displaystyle \ Delta \ Phi (\ mathbf {r}) = f (\ mathbf {r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/08b084144964fcc0684715b4d50f967cd693de5b)
![\ Phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
If it is known everywhere, the general solution of Poisson's equation, which approaches zero for large distances, is the integral
![{\ displaystyle f (\ mathbf {r})}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bb64b30ae67dec8ef9cb06c1d3537f00b9a7efed)
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.
In words: every charge at the location in the small area of size contributes additively to the potential at the location with its electrostatic or gravitational potential:
![{\ displaystyle \ mathrm {d} ^ {3} r '\, f (\ mathbf {r}')}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ea0b29cb19d70d91366978657a6257b92b92c83)
![\ mathbf r '](https://wikimedia.org/api/rest_v1/media/math/render/svg/1317b633a9366fab48e4b85ddec1cd1c0a8c31b3)
![{\ displaystyle \ mathrm {d} ^ {3} r '}](https://wikimedia.org/api/rest_v1/media/math/render/svg/bef930d406517d2df33110da85bb5f6d6c2c1b16)
![\ Phi](https://wikimedia.org/api/rest_v1/media/math/render/svg/aed80a2011a3912b028ba32a52dfa57165455f24)
![\ mathbf {r}](https://wikimedia.org/api/rest_v1/media/math/render/svg/eca0f46511c4c986c48b254073732c0bd98ae0c1)
![{\ displaystyle - {\ frac {\ mathrm {d} ^ {3} r '\, f (\ mathbf {r}')} {4 \, \ pi \, | \ mathbf {r} - \ mathbf {r } '|}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/c713a3321338e54535602c4d0a77b5f77431e2db)
Electrostatics
Since the electrostatic field is a conservative field , it can be expressed in terms of the gradient of a potential :
![\ Phi ({\ mathbf r})](https://wikimedia.org/api/rest_v1/media/math/render/svg/0aae204acbbf8529f6aebad36b0968d6eb4d8658)
![\ mathbf E (\ mathbf r) = - \ nabla \ Phi (\ mathbf r).](https://wikimedia.org/api/rest_v1/media/math/render/svg/4f237defb5df8e2518087651cfab91de491c4252)
Applying the divergence results in
![\ nabla \ cdot \ mathbf E (\ mathbf r) = - \ Delta \ Phi (\ mathbf r)](https://wikimedia.org/api/rest_v1/media/math/render/svg/6cac6f11f3dc715b027aa76ecc76a5734ac7204d)
with the Laplace operator .
![\Delta](https://wikimedia.org/api/rest_v1/media/math/render/svg/32769037c408874e1890f77554c65f39c523ebe2)
According to the first Maxwell equation , however, also applies
![\ nabla \ cdot \ mathbf E (\ mathbf r) = \ frac {\ rho (\ mathbf r)} {\ varepsilon}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6a00ec4e55c5b7ee703ef6c3721e84ab63314f11)
With
- the charge density
- the permittivity .
![\ varepsilon = \ varepsilon_ \ mathrm {r} \ cdot \ varepsilon_0](https://wikimedia.org/api/rest_v1/media/math/render/svg/c5981eb78d62a83a780ffe137c49ec7b9ba68cd8)
It follows for the Poisson equation of the electric field
![\ Delta \ Phi (\ mathbf r) = - \ frac {\ rho (\ mathbf r)} {\ varepsilon}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1a7629a539e556d5596c6fc37619141579169b11)
The special case for each location in the area under consideration is called the Laplace equation of electrostatics .
![{\ displaystyle \ rho (\ mathbf {r}) = 0}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9e4134ae2b6c8b5de205eeaad70897f533bb0eaf)
Electrodynamics of stationary currents
The emitter of a silicon solar cell , which can be described as purely two-dimensional to a good approximation, is considered here as an example . The emitter is located in the xy-plane, the z-axis points into the base. The lateral surface current density in the emitter depends on the z-component of the (volume) current density of the base occurring at the emitter , which is indicated by the continuity equation in the form
![{\ displaystyle \ mathbf {j} (x, y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2abd306f2206ac9cea4d0721e32b1e575056f12e)
![{\ displaystyle J_ {z} (x, y, z = 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/94bf4e22d41cb9ff5c167786cd2d137b01c959d4)
![{\ displaystyle \ nabla _ {2} \ cdot \ mathbf {j} (x, y) = - J_ {z} (x, y, z = 0)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/26f04c5fdbfa70a72e24d5e91a3c0c8069ff59f5)
can be described (with the two-dimensional Nabla operator ). The surface current density depends on the local Ohm's law with the lateral electric field in the emitter together: ; here is the specific sheet resistance of the emitter assumed to be homogeneous . If you write (as discussed in the section on electrostatics) the electric field as a gradient of the electric potential, you get a Poisson equation for the potential distribution in the emitter in the form
![{\ displaystyle \ nabla _ {2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/a5b7942a31d34704afaaf3b4ab68c434a6871b46)
![{\ displaystyle \ mathbf {j} (x, y) = R _ {\ Box} ^ {- 1} \ mathbf {E} (x, y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4cbc502c9ea71a6bfc3c2a081353b44898f30ce8)
![R _ {{\ Box}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/5ea3fff17b9b8208b36f229ce7b7a8cd045fde40)
![{\ displaystyle \ mathbf {E} (x, y) = - \ nabla _ {2} \ Phi (x, y)}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1b7074cd2b35bf060174580878f470c1f5873a42)
![{\ displaystyle \ Delta _ {2} \ Phi (x, y) = R _ {\ Box} J_ {z} (x, y, z = 0).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2e1961c0298995d8eaaa748b2651d7ca5888d2f4)
Gravity
Just like the electrostatic field
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the gravitational field g is also a conservative field:
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It is
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G is the gravitational constant
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the mass density.
Since only the charges are replaced by masses and by , analogously to the first Maxwell equation applies
![{\ displaystyle {\ frac {1} {4 \ pi \ varepsilon}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e73b22db97f1e7c1ada8304e8d3e5ff63e98672a)
![{\ displaystyle -G}](https://wikimedia.org/api/rest_v1/media/math/render/svg/9fb3535547044874239a93b1e765f0f71f7e610d)
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.
This results in the Poisson equation for gravity
![{\ displaystyle \ nabla \ mathbf {g} = - \ nabla \ cdot (\ nabla \ Phi _ {\ mathrm {m}} (\ mathbf {r})) = - 4 \ pi \ cdot G \ cdot \ rho _ {\ mathrm {m}} (\ mathbf {r}) \ Leftrightarrow}](https://wikimedia.org/api/rest_v1/media/math/render/svg/0971ec53216c7698eee9761decd3fe906a04152b)
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.
literature
Individual evidence
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↑ Wolfgang Nolting: Basic course in theoretical physics . [Online excl. the] 8th [dr.]. 3. Electrodynamics. Springer, Berlin, ISBN 978-3-540-71252-7 .