General Procedure for Calculating Electric Field of Distributed Charges Cut the charge distribution into pieces for which the field is known Write an expression for the electric field due to one piece (i) Choose origin (ii) Write an expression for E and its components Add up the contributions of all the pieces (i) Try to integrate symbolically (ii) If impossible – integrate numerically Check the results: (i) Direction (ii) Units (iii) Special cases

A Uniformly Charged Thin Ring Distance dependence: Far from the ring (z>>R): Ez~1/z2 Close to the ring (z<<R): Ez~z

θ dθ Q R Clicker Question A total charge Q is uniformly distributed over a half ring with radius R. The total charge inside a small element dθ is given by: A. B. C. D. E. D Choice One Choice Two Choice Three Choice Four Choice Five Choice Six

Clicker Question θ R Q +y dθ Q R +y Clicker Question A total charge Q is uniformly distributed over a half ring with radius R. The y component of electric field at the center created by a short element dθ is given by: A. B. C. D. B Choice One Choice Two Choice Three Choice Four

A Uniformly Charged Disk Along z axis Approximations: This is the result for disk on which has been placed a total charge Q uniformly distributed over its front and back surfaces. It is NOT a conducting disk. What is Ez for z>>R? Note: To a good degree of approximation when Z/R <<1, the field is independent of the distance from the disk. It is as if the disk is infinite in R. Close to the disk (0 < z < R) If z/R is extremely small Very close to disk (0 < z << R)

Field Far From the Disk Exact For z>>R Point Charge

Uniformly Charged Disk Edge On

Capacitor Two uniformly charged metal disks of radius R placed very near each other +Q -Q s A single metal disk cannot be uniformly charged: charges repel and concentrate at the edges Two disks of opposite charges, s<<R: charges distribute uniformly: Almost all the charge is nearly uniformly distributed on the inner surfaces of the disks; very little charge on the outer surfaces. We will use the previous result for disks made of insulating material. We will calculate E both inside and outside of the disk close to the center

Step 1: Cut Charge Distribution into Pieces We know the field for a single disk There are only 2 “pieces” +Q -Q s E+ Note that our diagram shows that each metal disk has almost the entire charge Q on the surface facing the other disk! This allows us to use the previous result to a good approximation. Treat each disk as if it had zero thickness. Enet E-

Step 2: Contribution of one Piece Origin: left disk, center Location of disks: z=0, z=s E- E+ Enet s Distance from disk to “2” z, (s-z) Left: Note that are making the approximation that all of the charge lies on the facing surfaces and thus the small amount of charge on the outside surfaces do not contribute to E. Right: z

Step 3: Add up Contributions Location: “2” (inside a capacitor)  Does not depend on z

Step 3: Add up Contributions Enet s z Location: “3” (fringe field) Note that we are dealing only with magnitudes of the electric fields here. Must take care when superimposing (summing) the contributions from the negative and positive plates. Note also that the fringe field points in a direction that would “discharge” the capacitor if connected in a circuit. That is, electrons would be drawn towards the positive plate. For s<<R: E1=E30 Fringe field is very small compared to the field inside the capacitor. Far from the capacitor (z>>R>>s): E1=E3~1/z3 (like dipole)

Electric Field of a Capacitor Enet s z Inside: Fringe: Step 4: check the results:  Units:

Which arrow best represents the field at the “X”? Clicker Question Which arrow best represents the field at the “X”? A) B) E C) E=0 D) E)

Electric Field of a Spherical Shell of Charge Field inside: Field outside: (like point charge)

E of a Sphere Outside  Direction: radial - due to the symmetry Divide into 6 areas: E1+E4 E2 E3 E6 E5

E of a Sphere Inside  Magnitude: E=0 Note: E is not always 0 inside – other charges in the Universe may make a nonzero electric field inside.

E of a Sphere Inside E=0: Implications Fill charged sphere with plastic. Will plastic be polarized? No! Solid metal sphere: since it is a conductor, any excess charges on the sphere arranges itself uniformly on the outer surface. There will be no field nor excess charges inside! Think of this in two steps. First, consider just the charged sphere. Inside, E=0. Introducing the (uncharged) plastic into the interior will not cause any polarization! In general: there is no electric field inside metals

Integrating Spherical Shell Divide shell into rings of charge, each delimited by the angle  and the angle + From ring to point: d=(r-Rcos) Surface area of ring: R  R Rsin  Rcos d r Q A mess of math

Exercise A solid metal ball bearing a charge –17 nC is located near a solid plastic ball bearing a uniformly distributed charge +7 nC (on surface). Show approximate charge distribution in each ball. Metal -17 nC Plastic +7 nC What is electric field field inside the metal ball?

Exercise Two uniformly charged thin plastic shells. Find electric field at 3, 7 and 10 cm from the center 3 cm: E=0 7 cm: 10 cm:

A Solid Sphere of Charge What if charges are distributed throughout an object? Step 1: Cut up the charge into shells R For each spherical shell: outside: r E inside: dE = 0 Outside a solid sphere of charge: for r>R

A Solid Sphere of Charge Inside a solid sphere of charge: R E r for r<R Why is E~r?  On surface:

Patterns of Fields in Space What is in the box? no charges? vertical charged plate?

Patterns of Fields in Space Box versus open surface …no clue… Seem to be able to tell if there are charges inside Gauss’s law: If we know the field distribution on closed surface we can tell what is inside.