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

3. The Continuity Equation for Steady State Currents Currents and current densities are constant in time – steady state. The flux of out of any closed surface must be zero.

4. Another form of Ohm’s Law

5. For steady state situation

6. Problem 4 Two wires having different resistivities ρ 1 and ρ 2 and equal cross sections, a, are connected end to end. Their lengths are l 1 and l 2. If a battery is connected to this system such that a potential difference of V is maintained between the ends, a)What will be the current densities in the wires? b) What will be the potential difference across each wire? c)Will there be any charge on the surface where the wires are connected?

7. 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?

8. Metal: ρ increases with increasing T Resistivity and temperature

9. Semiconductors: ρ decreases with increasing T

10. Superconductor 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 - Hg 2003

11. Water NitrogenOxygen 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

12. 1908- liquefied helium produced First discovered in mercury by Kamerlingh-Onnes in 1911. Critical temperature 4.21K. Nobel Prize in 1913. Superconductivity

13. High-T c 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

14. Meissner effect and magnetic levitation

15. 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.

16. Resistors in parallel: Resistors in series:

17. Circuit with capacitors