Patterns of Fields: Gauss’ Law, Ampere’s Law M&I Chapter 22

Electric Flux 电通量

One point charge

Gauss’ Law Coulomb’s Law VPython…

Using Gauss’ Law From flux to charge

How much charge is inside the cylinder ( 圆柱 )? E = 550 N/C r = 5 cm L = 15 cm q inside = 2.29 x C

Using Gauss’ Law From charge to the field

Uniformly charged planar surface

Uniformly charged spherical shell (outside)

Uniformly charged spherical shell (inside)

Uniformly charged cube Problem: The field is not uniform over each surface. So we can’t take E outside the flux integral.

Gauss’ Law is only useful when the field has a certain symmetry. Be careful: the law is always true, even when there is no symmetry.

-q+q Another example…

A charged conductor – where does the charge go? Gaussian surface Net charge is zero inside a conductor.

Excess charge in a conductor is always on the surface. Gaussian surface Net charge is zero inside a conductor.

What if there is a hole? Net charge is zero inside the Gaussian surface.

Could there be a field in the hole? The change in potential ΔV from A to B must be zero. So the field inside the hole must be zero.

No field in the hole.

Faraday cage

What if there is a charge in the hole? Net charge is still zero inside the Gaussian surface.

What if there is a charge in the hole? The net charge on the conductor is still Q, but some negative charge has moved to the surface of the hole.

Electric field just outside a conductor

Gauss’ Law for magnetic fields There are no magnetic “charges”.

Gauss’ Law for magnetic fields

Ampere’s Law

Is there a current passing through the inside of this path?

Ampere’s Law is a relationship between: the magnetic field around a closed path, and the net current passing through an open surface bounded by the path.

For a very long wire (L >> R):

Ampere’s Law

Ampere’s Law does not depend on the size and shape of the closed path, or of the surface bound by the path. It also does not depend on the position of the current inside the path. Like Gauss’ Law, we can use it to find the field in some situations with a high level of symmetry.

Magnetic field outside a cylindrical current-carrying conductor “Amperian loop”

Magnetic field inside a cylindrical current-carrying conductor “Amperian loop”

Magnetic field inside a cylindrical current-carrying conductor “Amperian loop” (direction from right-hand rule!)

Magnetic field inside a solenoid

N loops

Magnetic field inside a solenoid N loops

Maxwell’s Equations (so far…) *Not complete