10Summary Point Charges: Dipole: Line of Charge: Ring of Charge Coulomb’s LawElectric FieldThese can be used to find the fields in the vicinity of continuous charge distributions:+Dipole:Line of Charge:+Ring of ChargeCharged Plate:+++++
11From Chapter 23 – Coulomb’s Law We found the fields in the vicinity of continuous chargedistributions by integration:+Line of Charge:Are you interested in learning an easier way?Charged Plate:
12Electric flux The field is uniform The plane is perpendicular to the field
13Flux for a uniform electric field passing through an arbitrary plane The field is uniformThe plane is not perpendicular to the field
14Rules for drawing field lines The electric field, , is tangent to the field lines.The number of lines leaving/entering a charge is proportional to the charge.The number of lines passing through a unit area normal to the lines is proportional to the strength of the field in that region.Field lines must begin on positive charges (or from infinity) and end on negative charges (or at infinity). The test charge is positive by convention.No two field lines can cross.
15General flux definition The field is not uniformThe surface is not perpendicular to the fieldIf the surface is made up of amosaic of N little surfaces
16Electric Flux, GeneralIn the more general case, look at a small area elementIn general, this becomes
17Electric Flux Calculations The surface integral means the integral must be evaluated over the surface in questionIn general, the value of the flux will depend both on the field pattern and on the surfaceThe units of electric flux will be N.m2/C2
18Electric Flux, Closed Surface The vectors ΔAi point in different directionsAt each point, they are perpendicular to the surfaceBy convention, they point outward
20Flux from a point charge through a closed sphere
21Gauss’s Law Gauss asserts that the proceeding calculation for the flux from a point charge is true for any charge distribution!!!This is true so long as Q is the chargeenclosed by the surface of integration.
22A few questionsIf the electric field is zero for all points on the surface, is the electric flux through the surface zero?If the electric flux is zero, must the electric field vanish for all points on the surface?If the electric flux is zero for a closed surface, can there be charges inside the surface?What is the flux through the surface shown? Why?
23Example P24.1An electric field with a magnitude of 3.50 kN/C is applied along the x axis. Calculate the electric flux through a rectangular plane m wide and m long assuming thatthe plane is parallel to the yz plane;the plane is parallel to the xy plane;the plane contains the y axis, and its normal makes an angle of 40.0° with the x axis.
24Example P24.9The following charges are located inside a submarine: 5.00 μC, –9.00 μC, 27.0 μC, and –84.0 μC.Calculate the net electric flux through the hull of the submarine.Is the number of electric field lines leaving the submarine greater than, equal to, or less than the number entering it?
25Example P23.17A point charge Q = 5.00 μC is located at the center of a cube of edge L = m. In addition, six other identical point charges having q = –1.00 μC are positioned symmetrically around Q as shown in Figure P Determine the electric flux through one face of the cube.
26Applying Gauss’s Law Select a surface Try to imagine a surface where the electric field is constant everywhere. This is accomplished if the surface is equidistant from the charge.Try to find a surface such that the electric field and the normal to the surface are either perpendicular or parallel.Determine the charge inside the surfaceIf necessary, break the integral up into pieces and sum the results.e.g. For a cylinder,rh
27Example – a line of charge Find the correct closed surfaceFind the charge insidethat closed surface
34Example – a conducting conductor Insulators, like the previous charged sphere, trap excess charge so it cannot move.Conductors have free electrons not bound to any atom. The electrons are free to move about within the material. If excess charge is placed on a conductor, the charge winds up on the surface of the conductor. Why?The electric field inside a conductor is always zero.The electric field just outside a conductor is perpendicular to the conductors surface and has a magnitude, s/eo
35Einside = 0 Consider a conducting slab in an external field E If the field inside the conductor were not zero, free electrons in the conductor would experience an electrical forceThese electrons would accelerateThese electrons would not be in equilibriumTherefore, there cannot be a field inside the conductor
36Einside = 0, cont.Before the external field is applied, free electrons are distributed throughout the conductorWhen the external field is applied, the electrons redistribute until the magnitude of the internal field equals the magnitude of the external fieldThere is a net field of zero inside the conductorThis redistribution takes about 10-15s and can be considered instantaneous
37Charge Resides on the Surface Choose a gaussian surface inside but close to the actual surfaceThe electric field inside is zero (prop. 1)There is no net flux through the gaussian surfaceBecause the gaussian surface can be as close to the actual surface as desired, there can be no charge inside the surface
38Charge Resides on the Surface, cont Since no net charge can be inside the surface, any net charge must reside on the surfaceGauss’s law does not indicate the distribution of these charges, only that it must be on the surface of the conductor
39Field’s Magnitude and Direction Choose a cylinder as the gaussian surfaceThe field must be perpendicular to the surfaceIf there were a parallel component to E, charges would experience a force and accelerate along the surface and it would not be in equilibrium
40Field’s Magnitude and Direction, cont. The net flux through the gaussian surface is through only the flat face outside the conductorThe field here is perpendicular to the surfaceApplying Gauss’s law
41Conductors in Equilibrium, example The field lines are perpendicular to both conductorsThere are no field lines inside the cylinder
42Find the Electric Field: Example – a solid conducting sphere of charge surrounded by a conducting shell+2Q on inner sphere-Q on outer shellFind the Electric Field:cbarFind the correct closed surfaceFind the charge insidethat closed surface
43Example P24.31Consider a thin spherical shell of radius 14.0 cm with a total charge of 32.0 μC distributed uniformly on its surface. Find the electric field10.0 cm and20.0 cm from the center of the charge distribution.
44Example P24.35A uniformly charged, straight filament 7.00 m in length has a total positive charge of 2.00 μC. An uncharged cardboard cylinder 2.00 cm in length and 10.0 cm in radius surrounds the filament at its center, with the filament as the axis of the cylinder. Using reasonable approximations, find (a) the electric field at the surface of the cylinder and(b) the total electric flux through the cylinder.
45Example P24.43A square plate of copper with 50.0-cm sides has no net charge and is placed in a region of uniform electric field of 80.0 kN/C directed perpendicularly to the plate. Findthe charge density of each face of the plate andthe total charge on each face.
46Example P24.47A long, straight wire is surrounded by a hollow metal cylinder whose axis coincides with that of the wire. The wire has a charge per unit length of λ, and the cylinder has a net charge per unit length of 2λ. From this information, use Gauss’s law to find (a) the charge per unit length on the inner and outer surfaces of the cylinder and (b) the electric field outside the cylinder, a distance r from the axis.