Physics 2102 Gabriela Gonz lez Physics 2102 Electric fields Gauss law Carl Friedrich Gauss 1777 1855 r Q r l Q L l Q 2Rq l Q 2pR s Q pR2 E r R q q q r r E E kQ r2 kQ r r 2 L 2 2 k L r r 2 L 2 2 kQ sin 2k sin 2 R R E kQ 2k cos cos R2 R 2kQ r r E 2 1 2 1 2 2 2 R 2 r R r R 0 Electric field lines and forces We want to calculate electric fields because we want to predict how charges would move in space we want to know forces The drawings below represent electric field lines Draw vectors representing the electric force on an electron and on a proton at the positions shown disregarding forces between the electron and the proton e p e p e p e p d Imagine the electron proton pair is held at a distance by a rigid bar this is a model for a water molecule Can you predict how the dipole will move Electric charges and fields We work with two different kinds of problems easily confused Given certain electric charges we calculate the electric field produced by those charges Example we calculated the electric field produced by the two charges in a dipole Given an electric field we calculate the forces applied by this electric field on charges that come into the field Example forces on a single charge when immersed in the field of a dipole another example force on a dipole when immersed in a uniform field Electric Dipole in a Uniform Field Net force on dipole 0 center of mass stays where it is Net TORQUE t INTO page Dipole rotates to line up in direction of E t 2 QE a 2 sin q Qa E sinq p E sinq p x E The dipole tends to align itself with the field lines Distance between charges a Q Uniform Field E Q r p Potential energy of a dipole Work done by the field on the dipole When is the potential energy largest QE QE q r E Electric Flux Planar Surface Given planar surface area A uniform field E E makes angle q with NORMAL to plane Electric Flux F E A cos q Units Nm2 C Visualize flow of water through surface E q normal AREA A EA EA Electric Flux r r E dA Electric Flux A surface integral CLOSED surfaces define the vector dA as pointing OUTWARDS Inward E gives negative F Outward E gives positive F Electric Flux Example Closed cylinder of length L radius R Uniform E parallel to cylinder axis What is the total electric flux through surface of cylinder Note that E is NORMAL to both bottom and top cap E is PARALLEL to curved surface everywhere So F F1 F2 F3 pR2E 0 pR2E 0 Physical interpretation total inflow total outflow dA 1 2 3 dA dA Electric Flux Example Spherical surface of radius R 1m E is RADIALLY INWARDS and has EQUAL magnitude of 10 N C everywhere on surface What is the flux through the spherical surface a 4 3 pR2 E 13 33p Nm2 C b 4pR2 E 40p Nm2 C c 4pR2 E 40p Nm2 C What could produce such a field What is the flux if the sphere is not centered on the charge Gauss Law Consider any ARBITRARY CLOSED surface S NOTE this does NOT have to be a real physical object The TOTAL ELECTRIC FLUX through S is proportional to the TOTAL CHARGE ENCLOSED The results of a complicated integral is a very simple formula it avoids long calculations S Surface q E d A 0 One of Maxwell s 4 equations Gauss Law Example Infinite plane with uniform charge density s E is NORMAL to plane Construct Gaussian box as shown Applying Gauss law q A we have 2 AE 0 0 Solving for the electric field we get E 2 0 Two infinite planes E s 2e0 E s 2e0 Q Q E s e0 E 0 E 0 Gauss Law Example Cylindrical symmetry Charge of 10 C is uniformly spread over a line of length L 1 m Use Gauss Law to compute magnitude of E at a perpendicular distance of 1 mm from the center of the line Approximate as infinitely long line E radiates outwards Choose cylindrical surface of radius R length L co axial with line of charge E 1m R 1 mm Gauss Law cylindrical symmetry cont Approximate as infinitely long line E radiates outwards Choose cylindrical surface of radius R length L co axial with line of charge E 1m E A E 2 RL q L 0 0 L E 2k 2 0 RL 2 0 R R R 1 mm Compare with last class L 2 L 2 dx x E y k a 2 2 3 2 k a 2 2 2 a x a x a L 2 L 2 2k L 2 a 4a L if the line is infinitely long L a 2 k L2 k E y 2 aL a 2 Gauss Law Example Spherical symmetry q E d A 0 Consider a POINT charge q pretend that you don t know Coulomb s Law Surface Use Gauss Law to compute the electric field at a distance r from the charge r Use symmetry q draw a spherical surface of radius R E centered around the charge q E has same magnitude anywhere on surface E normal to surface E A E 4 r 2 q 0 F q e0 q kq E 2 2 2 A 4p r 4pe0 r r Gauss Law Example A spherical conducting shell has an excess charge of 10 C A point charge of 15 C is located at center of the sphere Use Gauss Law to calculate the charge on inner and outer surface of sphere a Inner 15 C outer 0 b Inner 0 outer 10 C c Inner 15 C outer 5 C R2 R1 15 C Gauss Law Example Inside a conductor E 0 under static equilibrium Otherwise electrons would keep moving Construct a Gaussian surface inside the metal as shown Does not have to be spherical Since E 0 inside the metal flux through this surface 0 Gauss Law says total charge enclosed 0 Charge on inner surface 15 C 5 C Since TOTAL charge on shell is 10 C Charge on outer surface 10 C 15 C 5 C 15C 15C Summary Gauss law F E dA provides a very direct way to compute the electric flux if we know the electric field In situations with symmetry knowing the flux allows us to compute the fields reasonably easily Electric field of a ring Let s calculate the field produced by a ring of radius R with total charge Q on a point on the axis at a distance z from the center A differential ring element will have charge dq and will produce a field dE with direction as shown in the figure The magnitude of the …
View Full Document
Unlocking...