Electric field of distributed charges

Electric field of a uniformly charged thin rod Before we start, think about the pattern of the electric field. What should it look like? What symmetry does it have? +

y x Magnitude: Unit vector (direction):

y x

y x Next step: Add up all the fields contributed by each piece (superposition principle)

y x Charge of each piece: “linear charge density”

y x Do the integration magic:

y x Result of the integration:

y r We can change x to r, since the field is rotationally symmetric:

y r What if we go very far away from the rod? Just like a point charge… makes sense!

y r What if we get very close to the rod (or have a very long rod)?

Steps for calculating electric field (1) Cut up the charge distribution into pieces. (2) Write an expression for the electric field due to one piece. (3) Add up the contributions of all the pieces (either as an integral, or with a computer). (4) Check the result.

y Electric field of a uniformly charged thin ring x z

Step 1: Cut it up into pieces.

Step 2: Write an expression for the field due to one piece.

Step 2: Write an expression for the field due to one piece.

Step 3: Add up all the contributions.

The total field points in the z-direction. Direction: parallel or anti-parallel to the axis, depending on the sign of Q.

The total field points in the z-direction.

The field away from the axis is more complicated…

Electric field of a uniformly charged disk Step 1: Cut it up into pieces. This time, each piece is a ring, of width Δr.

Electric field of a uniformly charged disk Step 2: Field due to a single piece.

Electric field of a uniformly charged disk Step 2: Field due to a single piece.

Electric field of a uniformly charged disk Step 3: Sum up all the contributions.

Electric field of a uniformly charged disk Step 3: Sum up all the contributions. For infinitesimally thin rings:

Electric field of a uniformly charged disk Step 3: Sum up all the contributions.

Electric field of a uniformly charged disk Step 3: Sum up all the contributions.

Electric field of a uniformly charged disk Very close to the disk: The field almost doesn’t change with distance, near the disk.

s -Q +Q Two uniformly charged disks: A capacitor Both fields are in the same direction (to the left).

s -Q +Q Two uniformly charged disks: A capacitor (assuming s << R) The field between the plates is approximately uniform.

- s R Two uniformly charged disks: A capacitor + + + + There is also a small field outside the plates: + + +

Electric field of a uniformly charged spherical shell This looks just like the field for a point charge!