Lectures 16, 17, 18 and 19

Question 1

What is another way of writing Greens Theorem?

Answer 1

double integral over R of (∇XF) dA

Question 2

If we want to determine how much rain falls through a closed loop, what factors effect it?

Answer 2

The area of the loop and the loops orientation.

Question 3

What is the equation for the total flux through a loop?

Answer 3

F.n(hat)*A

Question 4

What is the integral equation for the total flux?

Answer 4

Total flux = double integral over A of F.ds = double integral over A of F.n(hat) dA

Question 5

For a cartesian flat surface cube, what are the surface elements ds for each face?

Answer 5

Top face: ds = dxdy k(hat)

Side face: ds = dydz i(hat)

Question 6

For a cylindrical surface cylinder, what are the surface elements ds for each face?

Answer 6

Top Face: ds = ρdФdρ k(hat)
Barrel Face: ds = ρdФdz eρ k(hat)
Bottom Face: ds = -ρdФdρ k(hat)

Question 7

For a spherical surface, what is the surface element ds for the face?

Answer 7

ds = r^2 sinθ dθdФ er

Question 8

What integral do we need to evaluate for the total vector surface area?

Answer 8

s = double integral of ds

Question 9

What is an interesting part of the vector surface area for a 2D circle and a sphere with equal radius?

Answer 9

The vector surface area is the same, as both surfaces span the same boundary.

Question 10

What are two good example shapes for the vector surface area?

Answer 10

A scalar area of πa^2, and a hemisphere with scalar area 2πa^2.

Question 11

How do you calculate the vector surface area of the 2D circle?

Answer 11

Double integral of ds, where ds = dA k(hat), and dA = r drdФ. Do this integral and find s = πa^2 k(hat)

Question 12

How do you calculate the vector surface area for the hemisphere?

Answer 12

s = double integral of n(hat) dA. Have to consider 3 components for n(hat). Need to include a^2 sinθ in each integral, and just expand it out for each part, using the transformations. Find it also is s = πa^2 k(hat).

Question 13

Why isn’t the vector surface area the same for a hemisphere and a sphere?

Answer 13

The vector surface areas cancel out for the sphere, making it equal zero. The hemisphere is an open surface, whereas the sphere is a closed surface.

Question 14

What happens to the vector surface area for an overhang “doorknob” shape?

Answer 14

The vector surface areas cancel on each side of the overhangs, so s is unaffected.

Question 15

What is Stoke’s Theorem?

Answer 15

integral(closed) of F.dr = double integral over S of (∇XF).dS

Question 16

What do you do in the maths example for Stokes theorem with a square with one corner at the origin in the x-y plane with side a to work out the area? (F = x^2 i + x^2 j + x^2 k)

Answer 16

Find integral of F.dr: ds = dxdy k(hat), so (∇XF).dS = 2x dxdy. Take double integral and find F.dr = a^3

Question 17

What is the divergence theorem in 3 dimensions?

Answer 17

double integral over S of F.ds = triple integral over V of ∇.F dV

Question 18

What is the problem with using the 3D divergence theorem for an electric field around a sphere?

Answer 18

The E-field model blows up at r=0. We can normally only operate with well behaved functions.

Question 19

How do we solve the E-field problem at r=0?

Answer 19

To define ∇.E across all space, we use the dirac delta function, so ∇.E = Q/ε0 𝛿(r), and 𝛿(r) = 0 for r =/ 0

Question 20

How would you determine the electric field around a charged hollow sphere of radius a?

Answer 20

Use divergence theorem, E and ds are parallel.
Define ∇.E = σ/ε0 𝛿(r-a)
Do the integral and rearrange for E.
Find E = σ/ε0 a^2/r^2

Question 21

What is the divergence theorem for a scalar field?

Answer 21

double integral of Фds = triple integral of ∇Ф dV