Charge Distribution Running Head CHARGE DISTRIBUTION Charge Distribution in Hollow Spheres Pratibha Chopra Sukumaran PHY 690 Buffalo State College June 13 2008 Dr Dan MacIsaac 1 Charge Distribution 2 Charge Distribution in Hollow Spheres The direction and magnitude of electric field due to distribution of charges can be calculated The question is to estimate the charges knowing the 3D electric field pattern Gauss law relates pattern of electric field over a closed surface to the amount of charge inside the surface There are similarities between the Gauss Law and Coulombs law By adding up the electric fields due to all the individual electric charges q on a closed surface one can use Gauss law to find the amount and sign of the charge inside the closed surface Electric flux provides a quantitative relation between the amount and direction of electric field over an entire surface Electric field is considered positive when it leaves the surface and negative when it enters the surface and zero when it does not pierce the surface One can relate the direction and magnitude of electric flux to the angle the electric field E makes with the surface Field Due to a single point charge in space A point charge q1 creates an electric field that goes radially out II from the point throughout space r 1 q E1 4 0 r The field Charge Distribution 3 interac ts with anothe r charge q 2 wit ha r r 1 q1 q 2 r This is nothing but force F21 E1 q 2 2 4 0 r Coulomb s law of interaction between two charges The electric field is not absorbed by space and can come out from a positive charge and end at a negative charge This is analogous to flow of water from a sink to a drain The charge is conserved and lost on the way Electric field due to a point inside a hollow sphere A Next let us consider an enclosed point charge P near the inside P surface of the hollow sphere The charges to the upper left of the location P on the surface are closer to P Therefore each exerts a greater force to pull the charge towards the surface However there are many more charges farther away from P and this results in r E pointing to the right and downwards towards the surface r E Force and the q greater charge below balances the greater force per unit charge Charge on the inner surface is closer but less is effective According to Coulomb s law r E 1 r2 i e electric field due to the charge falls off rapidly as 1 r2 and results in the forces to the right and left cancelling each other out leading to a net zero charge inside the sphere At equilibrium reached in about 10 18 s mobile charges rearrange to give an internal charge of zero due to 1 r2 dependence Surface charge on the sphere is uniform and is to Surface area and spreads evenly on the surface The amount of electric field depends only on the amount of charge inside that area and not on the area This implies that r E coming out of the areas is the same and the electric field Charge Distribution 4 created by the charge is present throughout the space at all times This implies Einside 0 i e the electric field due to evenly spread charges on the surface is zero inside the shell because the only forces acting on the charge inside are due to charges on the surface of the sphere or outside the metal sphere B If we considered the charge enclosed in a sphere with a thick wall a hollow sphere as shown in Fig III the electric field coming out of the inner sphere will be the same as field coming out of the outer sphere This is essential as the charge has to be conserved The flow of P FIG III charge can be considered analogous to flow of water or fluidity which depends on how much water comes out of the spout and how much goes into the drain it is not dependent on the diameter of the pipe Similarly the charge or electric field coming out of the outer surface of the spherical object is the same as coming out of the inner sphere and is not dependent upon the size of the sphere It will be true of any shape A spherical surface is considered to make the math easier For a mathematical description Fig IVa and IVb are used to pictorially depict charge distribution on a surface due to a charge in a hollow sphere r E r A S Q IVa IVb E Charge Distribution Electric flux on a surface surface E r 5 A where A is a small area E the electric field and r is the unit vector This definition takes into account the direction and magnitude of the electric field and the surface area Electric field can be expressed as r E r dA when the surface is divided into infinitesimal areas A A clearer expression surface r E r dA to express the electric flux over the surface is S A spherical surface instead of any random shape is used to make the math easier and doesn t represent a spherical metallic surface vectors r E r E r cos E cos r xi yj zk The vector dot product of because the magnitude of r is 1 When the field points straight out of the surface as shown in Fig IVb i e when the E and r are to one another and are to the surface 0 is 0 at all locations of the surface Total field over the surface is E cos dA The surface area for the spherical surface is 4 r 2 total field on the surface is E 4 r 2 Since the total charge is conserved the charge on the surface is equal to charge inside i e E n A k Qinside Charge flux is closedsurface EA k r E dA Constant Qenclosed Q 4 r 2 Constant Q r2 The dynamic equilibrium inside a conductor is such that the charge is not defined over the entire space but is a function of position in the system and time To describe charge movement let us consider q space time or q r t where r is the position in the system or q x y z t At equilibrium the charge is on the outer sphere and inside q 0 Electric field can thus be expressed r E x y z t Consider a charge inside a conductor when equilibrium is not yet reached On a time scale of 10 18 s there is non uniform distribution of charge Fig I represents the 2D surface of the 3D space in which the charge is enclosed The surface is divided into infinitesimally small sections and each is labeled as dA The Charge Distribution 6 magnitude area of cross section whose direction is perpendicular to the section and pointing outward A point in space is sufficiently large …
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