Maxwell Equations
\(Div(F) =\nabla \cdot F\)
\(Curl(F) =\nabla \times F\)
1 Div
Assume given a vector field \(F\)
Positive = source
Negative = sink
- \(Div(F,(x,y))\) means Divergence of field F at point (x,y).
- Draw a circle at (x,y) in vector field
- Fluid moving out Circle > Fluid moving in Circle
- Positive
- Fluid moving in Circle > Fluit moving out Circle
- Negative
- Fluid moving out Circle > Fluid moving in Circle
Incompressible fluid like water has Divergence = 0 everywhere. Divergence = 0 implies no source or sinks.
2 Curl
How much fluid rotates around a point.
postive = Clockwise negative = Counterclockwise
Uni-directional vector field, imagine a vertical piece of wood, if vector field is stronger on top than bottom it will make it spin clockwise.
3 Maxwell Eq
\(Div(E) = \frac{\rho}{\epsilon_0}\)
- Divergence of electric field is correlated to charge density.
- Electric field has a source like effect when charge is positive.
- Electric field has a sink like effect when charge is negative.
\(Div(B) = 0\)
- magnetic field is incompressible like water
- magnetic field have no source nor sinks
- The north-south pole magnet effect is created a pair of clockwise and counter clockwise flow.
\(Curl(E) = -\frac{dB}{dt}\) A closed circuit denoted by the Curl of E will be induced a current when magnetic field changes. A magnetic field will be induced when current is running through a closed circuit.
\(Curl(B) = \mu_{0}(J + \epsilon_{0}\frac{dE}{dt})\)
4 Practical
A magnetic field is just an electric field from a different frame of reference.
Energy doesn’t JUST flow through wires in a circuit. The circuit has a electric-magnetic field that allows energy to flow in space and converge to the target.
