Current and Resistance Chapter 17 Current and Resistance

17.1 Electric Current Whenever electric charges move, an electric current is said to exist The current is the rate at which the charge flows through a certain cross-section For the current definition, we look at the charges flowing perpendicularly to a surface of area A

Definition of the current: Charge in motion through an area A. The time rate of the charge flow through A defines the current (=charges per time): I=DQ/Dt Units: C/s=As/s=A SI unit of the current: Ampere + -

Electric Current, cont The direction of current flow is the direction positive charge would flow This is known as conventional (technical) current flow, i.e., from plus (+) to minus (-) However, in a common conductor, such as copper, the current is due to the motion of the negatively charged electrons It is common to refer to a moving charge as a mobile charge carrier A charge carrier can be positive or negative

17.2 Current and Drift Speed Charged particles move through a conductor of cross-sectional area A n is the number of charge carriers per unit volume V (=“concentration”) nADx=nV is the total number of charge carriers in V

Current and Drift Speed, cont The total charge is the number of carriers times the charge per carrier, q (elementary charge) ΔQ = (nAΔx)q [unit: (1/m3)(m2 m)As=C] The drift speed, vd, is the speed at which the carriers move vd = Δx/Δt Rewritten: ΔQ = (nAvdΔt)q Finally, current, I = ΔQ/Δt = nqvdA Δx

Current and Drift Speed, final If the conductor is isolated, the electrons undergo (thermal) random motion When an electric field is set up in the conductor, it creates an electric force on the electrons and hence a current

Charge Carrier Motion in a Conductor The electric field force F imposes a drift on an electron’s random motion (106 m/s) in a conducting material. Without field the electron moves from P1 to P2. With an applied field the electron ends up at P2’; i.e., a distance vdDt from P2, where vd is the drift velocity (typically 10-4 m/s).

qvd (-q)(-vd) = qvd E vd E vd Does the direction of the current depend on the sign of the charge? No! qvd E vd (a) Positive charges moving in the same direction of the field produce the same positive current as (b) negative charges moving in the direction opposite to the field. E vd (-q)(-vd) = qvd

Current density: J=I/A= nqvdA/A=nqvd The current per unit cross-section is called the current density J: J=I/A= nqvdA/A=nqvd In general, a conductor may contain several different kinds of charged particles, concentrations, and drift velocities. Therefore, we can define a vector current density: J=n1q1vd1+n2q2vd2+… Since, the product qvd is for positive and negative charges in the direction of E, the vector current density J always points in the direction of the field E.

Example: An 18-gauge copper wire (diameter 1.02 mm) carries a constant current of 1.67 A to a 200 W lamp. The density of free electrons is 8.51028 per cubic meter. Find the magnitudes of (a) the current density and (b) the drift velocity.  

Solution: (a) A=d2p/4=(0.00102 m)2p/4=8.210-7 m2 J=I/A=1.67 A/(8.210-7 m2)=2.0106 A/m2 (b) From J=I/A=nqvd, it follows: vd=1.510-4 m/s=0.15 mm/s

17.3 Electrons in a Circuit The drift speed is much smaller than the average speed between collisions When a circuit is completed, the electric field travels with a speed close to the speed of light Although the drift speed is on the order of 10-4 m/s the effect of the electric field is felt on the order of 108 m/s

Meters in a Circuit – Ammeter An ammeter is used to measure current In line with the bulb, all the charge passing through the bulb also must pass through the meter (in series!)

Meters in a Circuit - Voltmeter A voltmeter is used to measure voltage (potential difference) Connects to the two ends of the bulb (parallel)

QUICK QUIZ 17.2 Look at the four “circuits” shown below and select those that will light the bulb.

17.4 Resistance and Ohm’s law In a homogeneous conductor, the current density is uniform over any cross section, and the electric field is constant along the length. b a V=Va-Vb=EL

Resistance The ratio of the potential drop to the current is called resistance of the segment: Unit: V/A=W (ohm)

Resistance, cont Units of resistance are ohms (Ω) 1 Ω = 1 V / A Resistance in a circuit arises due to collisions between the electrons carrying the current with the fixed atoms inside the conductor

V  I  V=const.I  V=RI Ohm’s Law Ohm’s Law is an empirical relationship that is valid only for certain materials Materials that obey Ohm’s Law are said to be ohmic I=V/R R, I0, open circuit; R0, I, short circuit

Ohm’s Law, final Ohmic Nonohmic Plots of V versus I for (a) ohmic and (b) nonohmic materials. The resistance R=V/I is independent of I for ohmic materials, as is indicated by the constant slope of the line in (a). Nonohmic