Current, Resistance, and Electromotive Force

Physics 231 Lecture 5-1 Fall 2008 Current

Current is the motion of any charge, positive or negative, from one point to another

Current is defined to be the amount of charge that passes a given point in a given amount of time dQ I = dt 1Coulomb Current has units of Ampere = 1sec

Physics 231 Lecture 5-2 Fall 2008 Drift Velocity Assume that an external electric field E has been established within a conductor Then any free charged particle in the r r conductor will experience a force given by F = qE The charged particle will experience frequent collisions, into random directions, with the particles compromising the bulk of the material

There will however be a net overall motion

Physics 231 Lecture 5-3 Fall 2008 Drift Velocity

There is net displacement given by vd∆t where vd is known as the drift velocity Physics 231 Lecture 5-4 Fall 2008 Drift Velocity

Consider a conducting wire of cross sectional area A having n free charge-carrying particles per unit volume with each particle having a charge q with

particle moving at vd The total charge moving past a given point is then given by

dQ = nqvd Adt the current is then given by dQ I = = nqv A dt d

Physics 231 Lecture 5-5 Fall 2008 Current Density

dQ This equation I = = nqv A dt d is still arbitrary because of the area still being in the equation We define the current density J to be

I J = = nqv A d

Physics 231 Lecture 5-6 Fall 2008 Current Density Current density can also be defined to be a vector r r J = nqvd Note that this vector definition gives the same direction for the current density whether we are using the positive or negative charges as the current carrier

Physics 231 Lecture 5-7 Fall 2008 Resistivity

The current density in a wire is not only dependent upon the external electric field that is imposed but It is also dependent upon the material that is being used Ohm found that J is proportional to E and in an idealized situation it is directly proportional to E

The resistivity is this proportionality constant and is given by E ρ = J The greater the resistivity for a given electric field, the smaller the current density

Physics 231 Lecture 5-8 Fall 2008 Resistivity The inverse of resistivity is defined to be the conductivity

The resistivity of a material is temperature dependent with the resistivity increasing as the temperature increases

This is due to the increased vibrational motion of the atoms the make up the lattice further inhibiting the motion of the charge carriers

The relationship between the resistivity and temperature is given approximately by

ρ(T )= ρ0[1+α(T −T0 )]

Physics 231 Lecture 5-9 Fall 2008 Resistivity

Let us take a length of conductor having a certain resistivity r r We have that E = ρ J But E and the length of the wire, L, are related to potential difference across the wire by V = E L I We also have that J = A Putting this all together, we then have V ρ I ρ L = or V = I L A A

Physics 231 Lecture 5-10 Fall 2008 Resistance

ρ L We take the last equation V = I A and rewrite it as V = I R ρ L with R = being the resistance A The resistance is proportional to the length of the material and inversely proportional to cross sectional area

V = I R is often referred to as Ohm’s Law

The unit for R is the ohm or Volt / Ampere

Physics 231 Lecture 5-11 Fall 2008 Example Two cylindrical resistors, R and R , are made 1 2 I1 I2 of identical material. R2 has twice the length of V R1 but half the radius of R1. These resistors are then connected to a battery V as shown:

What is the relation between I1, the current flowing in R1 , and I2 , the current flowing in R2?

(a) I1 < I2 (b) I1 = I2 (c) I1 > I2

The resistivity of both resistors is the same (r). Therefore the resistances are related as: L 2 2 L1 L1 R 2 = ρ = ρ = 8 ρ = 8 R 1 A 2 ( A1 / 4 ) A1 The resistors have the same voltage across them; therefore V V 1 I 2 = = = I1 R 2 8 R1 8 Physics 231 Lecture 5-12 Fall 2008 Resistance

Because the resistivity is temperature dependent, so is the resistance

R(T )= R0[1+α(T −T0 )]

This relationship really only holds if the the length and the cross sectional area of the material being used does not appreciably change with temperature

Physics 231 Lecture 5-13 Fall 2008 Electromotive Force A steady current will exist in a conductor only if it is part of a complete circuit

For an isolated conductor that has an external field impressed on it

Physics 231 Lecture 5-14 Fall 2008 Electromotive Force

To maintain a steady current in an external circuit we require the use of a source that supplies electrical energy

Whereas in the external circuit the current flows from higher potential to lower potential, in this source the current must flow from lower potential to higher potential, even though the electrostatic force within the source is in fact trying to do the opposite

In order to do this we must have an electromotive force, emf, within such a source

The unit for emf is also Volt

Physics 231 Lecture 5-15 Fall 2008 Electromotive Force

Ideally, such a source would have a constant potential difference, ε, between its terminals regardless of current

Real sources of emf have an internal resistance which has to be taken into account

The potential difference across the terminals of the source is then given by

Vab = ε − I rinternal

Physics 231 Lecture 5-16 Fall 2008 Energy

As a charge “moves” through a circuit, work is done that is equal to qVab This work does not result in an increase in the kinetic energy of the charge, because of the collisions that occur Instead, this energy is transferred to the circuit or circuit element within the complete circuit

Physics 231 Lecture 5-17 Fall 2008 Power

We usually are not interested in the amount of work done but in the rate at which work is done

This given by P =VabI If we have a pure resistance, we also have from before that

Vab = I R giving us the additional relations V 2 P = I 2R = ab R

Physics 231 Lecture 5-18 Fall 2008