Gauss’ Law for magnetic fields

There are no magnetic “charges”.

Gauss’ Law for magnetic fields

0 AdB

Ampere’s Law

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

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.

sdB

enclosedi

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

R

iB

2

40

RBdsBsdB 2

sdB R

R

i

2

2

40

i0Ri

idsB 0

Ampere’s Law

enclosed0isdB

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.

i

Magnetic field inside a cylindrical current-carrying conductor

r“Amperian loop”

R

B

rBsdB 2

2

2

enclosed R

rii

i

Magnetic field inside a cylindrical current-carrying conductor

r“Amperian loop”

R

B

rR

iB

20

2 (direction from

right-hand rule!)

i

Magnetic field outside a cylindrical current-carrying conductor

r

R

B

r

iB

20

B

Magnetic field inside a solenoid

Magnetic field inside a solenoid

Magnetic field inside a solenoid

B

BhsdB

nhii enclosed

h n loops per unit length

i

Magnetic field inside a solenoid

B

niB 0

h n loops per unit length

i

Maxwell’s Equations (so far…)

0

inside

q

AdE

0 AdB

enclosed0isdB

0 sdE

*Not complete

*Not complete