{-# OPTIONS_GHC -Wall #-} {-# LANGUAGE Trustworthy #-} {- | Module : Physics.Learn.Charge Copyright : (c) Scott N. Walck 2011-2014 License : BSD3 (see LICENSE) Maintainer : Scott N. Walck Stability : experimental This module contains functions for working with charge, electric field, electric flux, and electric potential. -} module Physics.Learn.Charge ( -- * Charge Charge , ChargeDistribution(..) , totalCharge -- * Electric Field , eField , eFieldFromPointCharge , eFieldFromLineCharge , eFieldFromSurfaceCharge , eFieldFromVolumeCharge -- * Electric Flux , electricFlux -- * Electric Potential , electricPotentialFromField , electricPotentialFromCharge ) where import Physics.Learn.CarrotVec ( magnitude , (*^) , (^/) ) import Physics.Learn.Position ( Position , ScalarField , VectorField , displacement , addFields ) import Physics.Learn.Curve ( Curve(..) , straightLine , simpleLineIntegral , dottedLineIntegral ) import Physics.Learn.Surface ( Surface(..) , surfaceIntegral , dottedSurfaceIntegral ) import Physics.Learn.Volume ( Volume(..) , volumeIntegral ) -- | Electric charge, in units of Coulombs (C) type Charge = Double -- | A charge distribution is a point charge, a line charge, a surface charge, -- a volume charge, or a combination of these. -- The 'ScalarField' describes a linear charge density, a surface charge density, -- or a volume charge density. data ChargeDistribution = PointCharge Charge Position -- ^ point charge | LineCharge ScalarField Curve -- ^ 'ScalarField' is linear charge density (C/m) | SurfaceCharge ScalarField Surface -- ^ 'ScalarField' is surface charge density (C/m^2) | VolumeCharge ScalarField Volume -- ^ 'ScalarField' is volume charge density (C/m^3) | MultipleCharges [ChargeDistribution] -- ^ combination of charge distributions -- | Total charge (in C) of a charge distribution. totalCharge :: ChargeDistribution -> Charge totalCharge (PointCharge q _) = q totalCharge (LineCharge lambda c) = simpleLineIntegral 1000 lambda c totalCharge (SurfaceCharge sigma s) = surfaceIntegral 100 100 sigma s totalCharge (VolumeCharge rho v) = volumeIntegral 50 50 50 rho v totalCharge (MultipleCharges ds) = sum [totalCharge d | d <- ds] {- shiftChargeDistribution :: Displacement -> ChargeDistribution -> ChargeDistribution shiftChargeDistribution d (Point -} -- | Electric field produced by a point charge. -- The function 'eField' calls this function -- to evaluate the electric field produced by a point charge. eFieldFromPointCharge :: Charge -- ^ charge (in Coulombs) -> Position -- ^ of point charge -> VectorField -- ^ electric field (in V/m) eFieldFromPointCharge q r' r = (k * q) *^ d ^/ magnitude d ** 3 where k = 9e9 -- 1 / (4 * pi * epsilon0) d = displacement r' r -- | Electric field produced by a line charge. -- The function 'eField' calls this function -- to evaluate the electric field produced by a line charge. eFieldFromLineCharge :: ScalarField -- ^ linear charge density lambda -> Curve -- ^ geometry of the line charge -> VectorField -- ^ electric field (in V/m) eFieldFromLineCharge lambda c r = k *^ simpleLineIntegral 1000 integrand c where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = lambda r' *^ d ^/ magnitude d ** 3 where d = displacement r' r -- | Electric field produced by a surface charge. -- The function 'eField' calls this function -- to evaluate the electric field produced by a surface charge. eFieldFromSurfaceCharge :: ScalarField -- ^ surface charge density sigma -> Surface -- ^ geometry of the surface charge -> VectorField -- ^ electric field (in V/m) eFieldFromSurfaceCharge sigma s r = k *^ surfaceIntegral 100 100 integrand s where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = sigma r' *^ d ^/ magnitude d ** 3 where d = displacement r' r -- | Electric field produced by a volume charge. -- The function 'eField' calls this function -- to evaluate the electric field produced by a volume charge. eFieldFromVolumeCharge :: ScalarField -- ^ volume charge density rho -> Volume -- ^ geometry of the volume charge -> VectorField -- ^ electric field (in V/m) eFieldFromVolumeCharge rho v r = k *^ volumeIntegral 50 50 50 integrand v where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = rho r' *^ d ^/ magnitude d ** 3 where d = displacement r' r -- | The electric field produced by a charge distribution. -- This is the simplest way to find the electric field, because it -- works for any charge distribution (point, line, surface, volume, or combination). eField :: ChargeDistribution -> VectorField eField (PointCharge q r') = eFieldFromPointCharge q r' eField (LineCharge lam c) = eFieldFromLineCharge lam c eField (SurfaceCharge sig s) = eFieldFromSurfaceCharge sig s eField (VolumeCharge rho v) = eFieldFromVolumeCharge rho v eField (MultipleCharges cds) = addFields \$ map eField cds ------------------- -- Electric Flux -- ------------------- -- | The electric flux through a surface produced by a charge distribution. electricFlux :: Surface -> ChargeDistribution -> Double electricFlux surf dist = dottedSurfaceIntegral 100 100 (eField dist) surf ------------------------ -- Electric Potential -- ------------------------ -- | Electric potential from electric field, given a position to be the zero -- of electric potential. electricPotentialFromField :: Position -- ^ position where electric potential is zero -> VectorField -- ^ electric field -> ScalarField -- ^ electric potential electricPotentialFromField base ef r = -dottedLineIntegral 1000 ef (straightLine base r) -- | Electric potential produced by a charge distribution. -- The position where the electric potential is zero is taken to be infinity. electricPotentialFromCharge :: ChargeDistribution -> ScalarField electricPotentialFromCharge (PointCharge q r') = ePotFromPointCharge q r' electricPotentialFromCharge (LineCharge lam c) = ePotFromLineCharge lam c electricPotentialFromCharge (SurfaceCharge sig s) = ePotFromSurfaceCharge sig s electricPotentialFromCharge (VolumeCharge rho v) = ePotFromVolumeCharge rho v electricPotentialFromCharge (MultipleCharges cds) = addFields \$ map electricPotentialFromCharge cds ePotFromPointCharge :: Charge -- ^ charge (in Coulombs) -> Position -- ^ of point charge -> ScalarField -- ^ electric potential ePotFromPointCharge q r' r = (k * q) / magnitude d where k = 9e9 -- 1 / (4 * pi * epsilon0) d = displacement r' r ePotFromLineCharge :: ScalarField -- ^ linear charge density lambda -> Curve -- ^ geometry of the line charge -> ScalarField -- ^ electric potential ePotFromLineCharge lambda c r = k *^ simpleLineIntegral 1000 integrand c where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = lambda r' / magnitude d where d = displacement r' r ePotFromSurfaceCharge :: ScalarField -- ^ surface charge density sigma -> Surface -- ^ geometry of the surface charge -> ScalarField -- ^ electric potential ePotFromSurfaceCharge sigma s r = k *^ surfaceIntegral 100 100 integrand s where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = sigma r' / magnitude d where d = displacement r' r ePotFromVolumeCharge :: ScalarField -- ^ volume charge density rho -> Volume -- ^ geometry of the volume charge -> ScalarField -- ^ electric potential ePotFromVolumeCharge rho v r = k *^ volumeIntegral 50 50 50 integrand v where k = 9e9 -- 1 / (4 * pi * epsilon0) integrand r' = rho r' / magnitude d where d = displacement r' r {- Student Exercise: Write a function for electric potential difference. -- | The electric potential difference V(end) - V(beginning) between the endpoints -- of a curve. electricPotentialDifference :: Curve -> ChargeDistribution -> Double electricPotentialDifference c dist = -dottedLineIntegral 1000 (eField dist) c -} --------------------------------- -- Common Charge Distributions -- ---------------------------------