static magnetic fields

Static magnetic fields Charge in motion yields a current I I = j  area j is a vector -- current density -- amperes/meter2 Ampere’s circuital law  B  dl = o Ienc

1 Tesla = 104 Gauss B at equator  1 Gauss

current ==> magnetic field

a I r  a a r B

a I r  a a r B

 B  dl = o Ienc due to one wire Ienc = I  B  dl = B [2 p r] B = o {1 / 2 p r} I due to other wire B = o {1 / 2 p r} I superposition

Ienc = N I  B  dl = B [2 L ] B = o N I / 2L B = o N I / L -- in center / top and bottom L

vector potential A we know that • B = 0 we know that • [ x vector] = 0 we can now specify the vector let vector be A such that B =  x A William Thomson shows that Neumann's electromagnetic potential A is in fact the vector potential from which may be obtained via B =  x A.

A and j are in the same direction!! vector potential A B =  x A we also know  x B = µo j  x  x A  = • A) - 2A - A  A = - µo j is similar to Poisson’s equation but we have to solve three PDE’s A and j are in the same direction!!

R z’ r dz’ 2 L I A z

R z’ r dz’ 2 L I B z after the integration

Biot-Savart integral R z’ r dz’ 2 L I B z

R z’ r dz’ 2 L I B z

magnetic dipole

large hadron collider

Earth’s magnetic field protects us

Inductance L

a b Coaxial cable

Inductance of a microstrip z d w B I

time-varying magnetic fields

An induced electric current flows in a direction such that the current opposes the change that induced it. This law was deduced in 1834 by the Russian physicist Heinrich Friedrich Emil Lenz (1804-65).

Faraday’s law either B or s individually change in time or they both change in time together

magnetic field changes in time x y z B ds a a magnetic field changes in time

size of loop changes in time x y z B ds size of loop changes in time

size of loop changes in time x y z L R w j B size of loop changes in time

Faraday’s law apply Stoke’s theorem

wire carrying current I L u DV I nonuniform B field

Bewley’s book trick questions not every motion generates a voltage uniform B & v substitution of circuit Vgen = 0!

X B 1 2 c u

B X 1 2 c u V12= 0

B X u c 1 2 V12= Bcu

B X u c 1 2 V12= Bcu

B X 1 2 c u V12= Bcu

B X u c 1 2 V12= Bcu V12= -Bcu

B X u c 1 2 V12= Bcu V12= -Bcu

I1 dl1 1 B1

I2 dl2 1 B2

I1 dl1 I2 dl2 1 B2 B1

I1 dl1 I2 dl2 1 B2 B1

I induced

electromagnetic launcher

J. Phys. D 33, 120 (2000)

before

before after