Electrical Current

So far we only have discussed charges that were not in motion -- electrostatics.

Now we turn to the study of charges in motion -- electrodynamics. When charges are in motion, then you have current.

Current in a wire In a time ∆t, a total charge ∆Q flows through the wire’s cross sectional area A. We define the current as I = ∆Q/∆t Units: Ampere = Coulomb/sec Drift Velocity: vd r As charges are now in motion, then the electric E ≠ 0 field is not zero inside the conductor.

But the charges in motion do not undergo a simple constant acceleration due to this field. The charges are constantly colliding with the atoms of the conducting material.

This endless cycle of acceleration and collision results in a net

“drift velocity” vd for the charges in motion. r E

vd Drift Velocity and Current # of charge carriers let n = (density of charge carriers) unit volume q = charge carried by each charge carrier A = wire cross sectional area

vd = drift velocity

then Avd = volume of charge carriers through A per second

nAvd = number of charge carriers through A per second and finally

I = qnAvd is the current in the wire Typical Value of Drift Velocity

Let’s say we have a wire 1 m long, and we apply a potential difference of 1 V from one end of the wire to the other. That means the electric field in the wire is 1 V/m. If an electron were freely accelerated from rest in that field, it would achieve an energy of 1 eV at the end of the wire, resulting in a speed of about 600,000 m/s.

Actual drift velocities are much slower than this! Consider the case of a copper wire of diameter 1.6 mm carrying a current of 1 A.

The charge carrier density in copper is related to the atom density. Each atom of copper contributes one

3 23 ρN ()8.93 g/cm ()6.02 ×10 28 3 electron for free conduction. Then n = A = Let’s follow the units= 8.47 to×10 electrons/m . M 63.5 g/mol

I 1 A −5 So, the drift velocity is vd = = = 3.67 ×10 m/s nqA ()8.47 ×1028 electrons/m3 ()1.6 ×10−19 C π()0.0016/2 m 2 So, it takes hours for an electron to travel from a light switch to a ceiling lamp. Resistance

The collisions of the charge carriers with the atoms in the material produce resistance to the flow of current. Typically, increasing the potential difference along the length of a wire produces an increase in current. We define resistance in terms of how much potential is required to produce a given amount of current.

V R = I

Volts units : Ohms ()Ω = Amps Ohm’s Law, and Ohmic Materials

In some materials, under certain circumstances, the resistance R is a constant -- it doesn’t depend on I or V. These materials obey Ohm’s Law and are said to be “ohmic.”

To learn whether a material is ohmic or R = constant not, you need to make a I = V/R measurement of current (I) as a function of R not constant applied potential (V) . ohmic non-ohmic Resistivity and Resistance

Resistivity is an intrinsic property of a material, like copper or iron or plastic. It is defined in terms of how much resistance a specific construction of the material would produce.

If we construct a wire of length L and cross sectional area A, then the L resistance of this particular wire and the resistivity of the material are related by R = ρ (L/A). A Units of resistivity: Ohm-meters

Resistivity is an intrinsic material property. Resistance is a property of a particular object. Temperature dependence of resistivity

Conducting materials display a slow change in resistivity as a function of temperature.

o ρ = ρ20[1+ α(Tc − 20 C )]

resistivity at 20 Co temperature in Co

temperature coefficient of resistivity Table of resistivities

⎧ ⎪ ⎪ ⎪ ⎪ ⎪ conductors ⎨ ⎪ ⎪ ⎪ Note how carbon has a ⎪ low resistivity (like a ⎩⎪ conductor) but a negative ⎧ temperature coefficient semi- conductors ⎨ ⎩ (like a semiconductor). ⎧ ⎪ ⎪ insulators ⎨ ⎪ ⎪ ⎩⎪ Table of wire gauges Resistors

A resistor is a component of an electrical circuit with a known value of resistance. It will obey Ohm’s law under some specific operating conditions.

R II

V When a current I flows through a resistor of resistance R, there is a drop in potential of V = IR. Table of resistor color code Energy consumption in electric circuits

We consider the movement of a volume of charge Q during a short time interval ∆t. During this time, a small amount of the charge, ∆Q, moves out of the volume at one end, and an equivalent amount moves in at the other end. The net effect is that a charge ∆Q moved from the higher potential Va to the lower potential Vb. If we call this potential drop V = Va− Vb, this charge experiences a drop in electric potential energy, ∆U=− ∆QV. Power needed to maintain a current

The rate of energy loss is ∆U/∆t=− (∆Q/∆t)V =−ΙV Since there was a drop in potential, there must have been some resistance in the current path. It is this resistance that eats up energy in an electric circuit.

The power needed to maintain current through a resistance is the product of the current through the resistance and the potential drop across the resistance: P = IV = I 2R ⎫ ⎬ power consumed by a resistance R = V 2 R⎭ R II

V