Electromagnetism

This will be the set of things I must rememeber

What is the del operator? What does it show?

Shows how quantities change in space

What three operations can be performed using the del operator?

Gradient, Divergence, Curl

What does gradient mean? Name one application and formula.

The direction and rate of greatest increase in a scalar field.

Eg. Temperature gradient points towards increasing temperature.

• Electric potential gradient relates to electric field.

What does divergence mean? Name one application and formula.

The amount a vector field spreads out from or converges into a point.

Eg. Positive divergence → source (fluid leaving)

• Negative divergence → sink (fluid entering)

What does curl mean? Name one application and formula.

The local rotation or “twisting” of a vector field.

Imagine placing a tiny paddle wheel in a fluid:

• spinning → curl exists

• not spinning → curl = 0

Curl inputs a ____ field and outputs a ___ field.

Vector, vector

Divergence inputs a _____ field and outputs a _____ field.

Vector, scalar

Gradient inputs a _____ field and outputs a _____ field.

Scalar, vector

What is the Laplacian? Formula? Application?

Tells you whether a point is higher or lower than the average of the points around it. - The divergence of the gradient.

Eg. The Laplacian tells you where heat will flow.

Example:

• A hot spot has TTT higher than its surroundings.

• Laplacian < 0

• Heat spreads away until the temperature evens out.

For divergence-less (aka_____) fields, nabla.F = ______ , closed integral of F . da = ____ for _____, the integral of F . da is ____ for ______, F = nabla x A is a ________ and magnetic field B is _____

solenoidal, 0 everywhere, 0 for any closed surface, independence of surface for any given boundary line, curl of a vector, divergence-less

For curl-less (irrotational) fields, nabla x F = ___ ____, the closed integral of F . dl = ___ for ______, the integral from b to a of F . dl is ________ , F = -nabla V is a _________

0 everywhere, 0 for any closed path, independent of path, a gradient of a scalar

Helmholtz Theorem - Formula and meaning

Any sufficiently smooth vector field can be split into two independent parts, the curl-less part and the divergence-free part.

The Integral of the Dirac Delta Functions is equal to ___, at x = 0 it is ___ and ____ everywhere else

1, infinity, 0

What is Stokes Theorem? What does it represent?

Relates the surface integral of a curl to the line integral around the boundary. (3D)

What is Green’s Theorem? What does it represent?

The total rotation (curl) inside a region equals the circulation around its boundary. (2D)

What is Gauss’s Theorem? What does it represent?

Outflow through the surface equals the amount generated inside.