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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