General Procedure for Calculating Electric Field of Distributed

General Procedure for Calculating Electric Field of Distributed Charges 1. Cut the charge distribution into pieces for which the field is known 2. Write an expression for the electric field due to one piece (i) Choose origin (ii) Write an expression for E and its components 3. Add up the contributions of all the pieces (i) Try to integrate symbolically (ii) If impossible – integrate numerically 4. 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/z 2 Close to the ring (z<<R): Ez~z

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. E. D. 1. 2. 3. 4. 5. 6. Choice One Choice Two Choice Three Choice Four Choice Five Choice Six dθ R C. Q θ

+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: 1. 2. 3. 4. A. B. C. D. Choice One Choice Two Choice Three Choice Four dθ R Q θ

A Uniformly Charged Disk Along z axis Approximations: Close to the disk (0 < z < R) Very close to disk (0 < z << R) If z/R is extremely small

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 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. s 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 Enet +Q E+ s E-

Step 2: Contribution of one Piece Origin: left disk, center Location of disks: z=0, z=s Distance from disk to “ 2” z, (s-z) E+ Enet Left: E- Right: s 0 z

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

Step 3: Add up Contributions Location: “ 3” (fringe field) Enet E+ Es 0 z For s<<R: E 1=E 3 0 Fringe field is very small compared to the field inside the capacitor. Far from the capacitor (z>>R>>s): E 1=E 3~1/z 3 (like dipole)

Electric Field of a Capacitor Inside: Enet Fringe: E+ E- s Step 4: check the results: Units: 0 z

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

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: E 6 E 5 E 2 E 1+E 4 E 3

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. No! Will plastic be polarized? 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: there is no electric field inside metals

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

Exercise A solid metal ball bearing a charge – 17 n. C is located near a solid plastic ball bearing a uniformly distributed charge +7 n. C (on surface). Show approximate charge distribution in each ball. Metal -17 n. C Plastic +7 n. C What is electric 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: 7 cm: 10 cm: E=0

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

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

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

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