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

Capacitor Two uniformly charged metal disks of radius R placed very near each other +Q -Q s 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 Why must there be charge on the outer surfaces?

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) E- E+ Enet s z  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:

Exercise Given: capacitor, radius R=50 cm, gap s=1 mm (air). Find: maximum charge before sparks are formed (Ecrit=3106 N/C) Solution: What Q would cause sparking if spacing s  2s? What is the attractive force between the plates? The second equation is correct since the electric field of one plate is Ecrit/2 and it acts on the Q of the other plate. F=QE= (2.110-5C)(3106 N/C)=63 N F=Q(Ecrit/2)= (2.110-5C)(3106/2 N/C)=31.5 N

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

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 Outside  Magnitude: As long as we are far from a region of distributed charges we can approximate the electric field of that region as being due to a point charge. How would a charged sphere interact with other charges? - as a point charge (same force)

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! In general: In static equilibrium, there is no electric field inside metals

E of a Sphere Inside What is electric field right at the surface? Electric field at the surface is highly variable in magnitude and direction Need to be >1000 atomic diameters away from surface for equations to work!

Integration Divide shell into rings of charge, each delimited by the angle  and the angle + Use polar coordinates (r, ,). Distance from center: d=(r-Rcos) Surface area of ring: R  R Rsin  Rcos d r

Question 2 (Chap. 16) Which one of these statements is false? The electric field of a very long uniformly charged rod has a 1/r distance dependence. The electric field of a capacitor at a location outside the capacitor is very small compared to the field inside the capacitor. The fringe field of a capacitor at a location far away from the capacitor looks like an electric field of a point charge. The electric field of a uniformly charged thin ring at the center of the ring is zero. Answer: C It looks like a dipole field!

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:

Exercise 1 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. Note that the fact that the field inside the metal sphere is zero results from the superposition of the field due to the non-uniform negative surface charge distribution on the metal sphere and the field due to the uniformly distributed positive charge on the plastic sphere. Note also that the interior of the plastic ball becomes polarized due to the presence of the metal ball, and not because of the uniform positive charge on the plastic ball. Metal -17 nC Plastic +7 nC What is electric field field inside the metal ball?

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)

Announcements – 272H EXAM 1 – Thursday, Feb. 13, 8-9:30 pm in room 203 Chapters 14, 15, & 16 Equation sheet will be provided Pencil, calculator There will be no lecture on Monday Feb. 17