Short Version : 21. Gauss’s Law

21.1. Electric Field Lines Electric field lines = Continuous lines whose tangent is everywhere // E. They begin at + charges & end at  charges or . Their density is  field strength or charge magnitude. Vector gives E at point Field line gives direction of E Spacing gives magnitude of E

Field Lines of Electric Dipole Direction of net field tangent to field line Field is strong where lines are dense.

Field Lines

21.2. Electric Flux & Field 8 lines out of surfaces 1, 2, & 3. But 2  2 = 0 out of 4 (2 out, 2 in). 16 lines out of surfaces 1, 2, & 3. But 0 out of 4.  8 lines out of (8 into) surfaces 1, 2, & 3. But 0 out of 4.

Number of field lines out of a closed surface  net charge enclosed. 8 lines out of surfaces 1 & lines out of surface 3. 0 out of 4. 8 lines out of surface 1. 8 lines out of surface 2. 4  4 = 0 lines out of surface 3. Count these. 1:  4 2: 8 3: 4 4: 0

Electric Flux Electric flux through flat surface A : A flat surface is represented by a vector where A = area of surface and [  ]  N m 2 / C. Open surface: can get from 1 side to the other w/o crossing surface. Direction of A ambiguous. Closed surface: can’t get from 1 side to the other w/o crossing surface. A defined to point outward. E ,  A , 

21.3. Gauss’s Law Gauss’s law: The electric flux through any closed surface is proportional to the net charges enclosed.  depends on units. For point charge enclosed by a sphere centered on it: SI units  = vacuum permittivity Gauss’s law: Field of point charge:

Gauss & Coulomb For a point charge: E  r 2E  r 2 A  r 2   indep of r. Gauss’ & Colomb’s laws are both expression of the inverse square law. Principle of superposition  argument holds for all charge distributions For a given set of field lines going out of / into a point charge, inverse square law  density of field lines  E in 3-D. Outer sphere has 4 times area. But E is 4 times weaker. So  is the same

21.4. Using Gauss’s Law Useful only for symmetric charge distributions. Spherical symmetry:( point of symmetry at origin ) 

Example Uniformily Charged Sphere A charge Q is spreaded uniformily throughout a sphere of radius R. Find the electric field at all points, first inside and then outside the sphere.   True for arbitrary spherical  (r).

Example Hollow Spherical Shell A thin, hollow spherical shell of radius R contains a total charge of Q. distributed uniformly over its surface. Find the electric field both inside and outside the sphere.  Contributions from A & B cancel. Reflection symmetry  E is radial.

Example Point Charge Within a Shell A positive point charge +q is at the center of a spherical shell of radius R carrying charge  2q, distributed uniformly over its surface. Find the field strength both inside and outside the shell. 

Tip: Symmetry Matters Spherical charge distribution inside a spherical shell is zero  E = 0 inside shell E  0 if either shell or distribution is not spherical. Q = q  q = 0 But E  0 on or inside surface

Line Symmetry Line symmetry:  r  = perpendicular distance to the symm. axis. Distribution is independent of r //  it must extend to infinity along symm. axis.

Example Infinite Line of Charge Use Gauss’ law to find the electric field of an infinite line charge carrying charge density in C/m.   c.f. Eg True outside arbitrary radial  (r  ). (radial field) No flux thru ends

Example A Hollow Pipe A thin-walled pipe 3.0 m long & 2.0 cm in radius carries a net charge q = 5.7  C distributed uniformly over its surface. Fine the electric field both 1.0 cm & 3.0 cm from the pipe axis, far from either end.  at r = 1.0 cm at r = 3.0 cm

Plane Symmetry Plane symmetry:  r  = perpendicular distance to the symm. plane. Distribution is independent of r //  it must extend to infinity in symm. plane.

Example A Sheet of Charge An infinite sheet of charge carries uniform surface charge density  in C/m 2. Find the resulting electric field.   E > 0 if it points away from sheet.

21.5. Fields of Arbitrary Charge Distributions Dipole : E  r  3 Point charge : E  r  2 Line charge : E  r  1 Surface charge : E  const

Conceptual Example Charged Disk Sketch some electric field lines for a uniformly charged disk, starting at the disk and extending out to several disk diameters. point-charge-like infinite-plane-charge-like

Making the Connection Suppose the disk is 1.0 cm in diameter& carries charge 20 nC spread uniformly over its surface. Find the electric field strength (a) 1.0 mm from the disk surface and (b) 1.0 m from the disk. (a) Close to disk : (b) Far from disk :

21.6. Gauss’s Law & Conductors Electrostatic Equilibrium Conductor = material with free charges E.g., free electrons in metals. External E  Polarization  Internal E Total E = 0 : Electrostatic equilibrium ( All charges stationary ) Microscopic view: replace above with averaged values. Neutral conductorUniform field Induced polarization cancels field inside Net field

Charged Conductors Excess charges in conductor tend to keep away from each other  they stay at the surface. More rigorously: Gauss’ law with E = 0 inside conductor  q enclosed = 0  For a conductor in electrostatic equilibrium, all charges are on the surface.  = 0 thru this surface

Example A Hollow Conductor An irregularly shaped conductor has a hollow cavity. The conductor itself carries a net charge of 1  C, and there’s a 2  C point charge inside the cavity. Find the net charge on the cavity wall & on the outer surface of the conductor, assuming electrostatic equilibrium. E = 0 inside conductor   = 0 through dotted surface  q enclosed = 0  Net charge on the cavity wall q in =  2  C Net charge in conductor = 1  C = q out + q in  charge on outer surface of the conductor q out = +3  C q out +2  C q in

Experimental Tests of Gauss’ Law Measuring charge on ball is equivalent to testing the inverse square law. The exponent 2 was found to be accurate to 10  16.

Field at a Conductor Surface At static equilibrium, E = 0 inside conductor, E = E  at surface of conductor. Gauss’ law applied to pillbox surface:  The local character of E ~  is incidental. E always dependent on ALL the charges present.

Dilemma? E outside charged sheet of charge density  was found to be E just outside conductor of surface charge density  is What gives? Resolution: There’re 2 surfaces on the conductor plate. The surface charge density on either surface is . Each surface is a charge sheet giving E =  /2  0. Fields inside the conductor cancel, while those outside reinforce. Hence, outside the conductor

Application: Shielding & Lightning Safety Coaxial cable Strictly speaking, Gauss law applies only to static E. However, e in metal can respond so quickly that high frequency EM field ( radio, TV, MW ) can also be blocked (skin effect). Car hit by lightning, driver inside unharmed.

Plate Capacitor inside outside Charge on inner surfaces only