Week 4

Question 1

2 goal equations of electrostatics

Answer 1

Question 2

Harmonic functions

Answer 2

Solutions to laplaces equation

(Not only examples)

Question 3

form integral form of Gauss’s law for a region of zero net charge in terms of φ

Answer 3

where we use E = grad*phi

Question 4

Greene’s second identity

Answer 4

Question 5

Apply greens second identity and laplaces equation to a region V which is a ball radius R and for f = 1/r g = φ

Answer 5

Question 6

Key insight of harmonic functions

Answer 6

They are averaging functions and have no local extrema across V. Any extrema must occur at boundary of V

If φ satisfies Laplace’s equation in a region V and in addition is constant on boundary of V then φ must be equal to the same value across the entire region as the boundary

Question 7

Dirichlet boundary conditions

Answer 7

Values of a function itself on a boundary (for PDE)

Question 8

Neumann boundary conditions

Answer 8

Values of normal derivative of function along a boundary (for PDE)

Question 9

Showing uniqueness of solution to Laplace’s equation in some volume V

Answer 9

1) let there be 2 solutions
2) construct a difference solution
3) on boundary all solutions must be equal
4) therefore both solutions equal

If Φ is specified on the boundary of V, S

Then below:

Question 10

Integration by parts

Answer 10

Question 11

Taking inner product for sin(nx)

Answer 11

Question 12

Requirements for stable equilibrium

Answer 12

• force in a positive test charge Q is zero at the equilibrium point r0 (E(r0) = 0)
• when displaced from the equilibrium point forces act so as to return the test charge Q to r0
  (Field lines all point towards r0)

Question 13

For a charge q located at the origin ρ(r) = ?

Answer 13

q * δ³(r) which of course spikes at origin

Question 14

In spherical coordinates, how to ‘remove’ laplacians from equation

Answer 14

Also grad(1/r) = 1/r * e_r

Question 15

Relate Laplace’s equation to Poisson’s

Answer 15

Laplace’s equation is a special cause of Poisson’s used in regions where ρ = 0

Question 16

How to go from E to Φ

Answer 16

You may need to split the problem

Take line integral

Question 17

Uniformly charged sphere has charge inside? Outside ?

Answer 17

Inside: zero it’s a closed surface therefore there is no net flux and the charges are distributed symmetrically around the surface

Outside: as the sphere act like a point charge at its centre use the expression for a point charge

Question 18

What is grounding

Answer 18

Potential is zero on the surface

Question 19

Method of images: why? How?

Answer 19

Used in problems where we’re looking for field or potential but conductors make it difficult to do so

Additional Charges are chosen to mimic conductors zero potential

Question 20

Separation of variables

Answer 20

Generally:
1) boundary conditions of potential
2) define variable separated potential
3) sub into Laplace’s eq and divide by variable separated Φ
4) solve individual equations with boundaries
5) determine which functions it must be

Question 21

Green’s function

Answer 21

Question 22

Using green’s function to solve Poisson’s equation

Answer 22

Question 23

Use the Heaviside to construct a finite length between -d/2 and d/2

Answer 23