Current Density

S

Sdji

Consider current flowing in a homogeneous wire with cross sectional area A.

jAdSjjdSSdjiA A A

A

ij

The Continuity Equation for Steady State Currents

0Sdj

Currents and current densities are constant in time – steady state. The flux of out of any closed surface must be zero.

j

Another form of Ohm’s Law

Ej

jE

For steady state situation

0 Sdj

0 rdE

Problem 4

Two wires having different resistivities ρ1 and ρ2 and equal cross sections, a, are connected end to end. Their lengths are l1 and l2. If a battery is connected to this system such that a potential difference of V is maintained between the ends,

• What will be the current densities in the wires?• What will be the potential difference across each wire?• Will there be any charge on the surface where the wires

are connected?

Exercise 5

Consider a cylindrical shell, inner radius a and outer radius b. It is made of material with resistivity ρ. Suppose a current can be made to flow out from the inner surface to the outer. What would the resistance be for this current?

Metal: ρ increases with increasing T

T

Resistivity and temperature

Semiconductors: ρ decreases with increasing T

T

Superconductor

T

0cT

Once a current has been established in a superconducting ring, it continues indefinitely without the presence of any driving field.

1911 Dutch physicist Heike Kamerlingh Onnes - HgKTc 2.4

2003 KTc 160

Water Nitrogen Oxygen

Boils

Freezes

100 C (212 F) -196 C (-322 F) -183 C (-297 F)

0 C (32 F) -210 C (-346 F) -223 C (-369 F)

Our air is ¾ Nitrogen and ¼ Oxygen

• 1908- liquefied helium produced

• First discovered in mercury by Kamerlingh-Onnes in 1911.

• Critical temperature 4.21K.

• Nobel Prize in 1913.

Superconductivity

High-Tc Superconductivity

Liquid nitrogen temperature 77 K

Complex ceramic materials were discovered in 1986. They exhibit superconductivity at much higher temperatures – above LN temperature!

Muller and Bednortz, Nobel Prize 1987

Meissner effect and magnetic levitation

For steady state situation

1.Kirchhoff’s junction rule: The algebraic sum of the currents into any junction is zero.

2.Kirchhoff’s loop rule: The algebraic sum of the potential differences in any loop must be zero.

0 Sdj

0 rdE

Resistors in parallel:

Resistors in series:

21 RRR

21

111

RRR

Circuit with capacitors