Chapter 27: Current and Resistance

Chapter 27: Current and Resistance introduces the concept of Current Density (J). We are provided a series of equations that compliment our understanding of "what is current density?"  "How do I utilize this concept to better understand the world?"

Current Density is a vector quantity that is defined as: J Units for J: Amps/m^2J points in the direction CURRENT IS FLOWING/MOVING.
J is a function of time and space.

Current Density (J) of an ohmic conductor is proportional to the electric field WHEN FLOW IS UNIFORM:

• J = (conductivity of the material)(Electric field)

NOTE: conductivity is not a function of E

Wait ... what is conductivity?

Google Definition says:
Conductivity is the degree to which a specified material conducts electricity, calculated as the ratio of the current density in the material to the electric field that causes the flow of current. It is the reciprocal of the resistivity.

therefore......

•  in the absence of an electric field, the average velocity of an electron equals zero
• current density is dependent on the electric field

This brings up the topic of OHMIC MATERIALS:

An ohmic material is a material in which the conductivity of the material is not a function of the electric field.

OHMS LAW

where V is the potential difference, aka: voltage difference, delta V, voltage change

*NOTE: when voltage increases, current increases!

Now, back to the topic of current density....

WHEN FLOW IS NOT UNIFORM, what is the current density?

NOTE: J is perpendicular to the surface
NOTE: J and dA are always paralell
NOTE: I = int[J(A)dA]

Additional useful equations from this topic:

The average current in a conductor is related to the motion of the charge carriers:

• Iavg = nqvA

n= density of charge carriers
q = charge on each carrier
v = drift speed
A = xsection areaof the conductor

The resistance over length "l" for a uniform block with a xsection area "A":

• R = (rho)[(l/A)]

rho = resistivity of the material (given in a table, or google it!)

resistance is equal to the resistivity constant, rho: p multiplied by the length of the object, all divided by the area of the object:

Resistivity of a conductor varies linearly with temperature according to this equation: 

• rho = rhoo[(1+alpha(T-To)]

Questions to think about:

Is RESISTANCE the same as RESISTIVITY? 

NO: resistivity is an intrinsic material property that is found on a table (or on google....) 

How is conductivity related to RESISTIVITY?

Conductivity is inversely proportional to resistivity. when conductivity goes down, resistivity goes up!

METALS: have high conductivity, low resistivity

INSULATORS: have low conductivity, high resistivity 

Class Practice from lecture*: 10-7-13

1. Given the VAVG, q, A, n find the current (I).

total # charges = n(volume) = n(area)
               
             total # electrons that cross = n(A)(VAVG)(T2-T1)

Q2-Q1 = total charge = (#)(q)
Q2-Q1 = total charge = [n(A)(VAVG)(T2-T1)][q]
                 
I = Q2-Q1 / T2-T1

I = [n(A)(VAVG)(T2-T1)](q)] / (T2-T1)
I = nA(VAVG)q

2. Calculate the total amount of blood flow **friction is involved!!**

*note: when R = 0, flow is at a maximum
*note: when r = R flow is at a minimum

 - find the current: 

I = integral[J(A)(d(a)(cos(0))] 

I = integral[J(R)2pi(r)dr] from 0 -> R

I = integral[Ji - (Ji/R)^r(2pi(r)dr)] from 0 -> R

I = integral[Ji 2pi(1-(1/R)^r)dr] from 0 -> R

I = 2piJi [(r^2/2)-(r^3/3R)]^r from 0 -> R

3. In a wire, J is not dependent upon dA. Give the equation for I in terms in J and A. **NO Friction!**

I = (Q2-Q1)/(T2-T1) 
I = integral[J dot dA] 
I = (J)integral[dA]
I = JA

for wires:
Current = I = JA
                J = I/A
4. Given that I = (n)(q)VAVG(A), find an expression for J in a wire:

J = (n)(q)(VAVG)
where n = charge density 


*(P. Francis, De Anza College, Physics 4B lecture on chapter 27)

ELECTRICAL POWER: "the rate of doing work"

P = power of the battery
Unit for Power: Joules/second = Watt

Power: is the rate at which energy us being converted from chemical to electric potential energy & thermal energy (heat loss).

• PBATT = V(dQ/dt) = (V)(I)

                                                               memorization tip: P=IV  can be pronounced: "peev"

* for an OHMIC material:

• PBATT = (I)^2(R)

where
 I(t) is a function of time

If a potential difference is maintained across a circus element, the power (aka rate of energy supplied to the element) is:

• Power = I(Vf-Vi)

Power delivered to a resistor: 

• P = (I)^2R = (Vf-Vi)^2 / R

Why can we describe the power delivered to a resister with this equation?
Answer: because the potential difference across a resister is (Vf-Vi) = IR

Work done by power: 
from t=o to t=end



Example from lecture (P. Francis, De Anza College, Physics 4B)

Calculate the resistance of the right 1/3 of the material shaped as a cone:

step 1: find A(x)

A(x) = pi(r)^2
where r = (slope)x+b
where b =0
r = (slope)x
r = (R/H)x

step 2: integrate

R = int[p(dx)/[pi(R^2/H^2)x^2]] from 2/3H to 1H

Additional equations introduced in this chapter: 

When an electric field is applied the average velocity of electrons is equal to the Drift Velocity vd

• vd = [(q)(E)/me]T

T = average time interval between electron-atom collisions
me = mass of the electron

According to this model, the resistivity of metal is:

• rho = [me / (n(q)^2T)]