Electric Current


In this chapter we will be dealing with electric current. Electric Current is something that everyone encounters on a daily basis, and can be both a help and a danger to an individual. Without current we would not be able to start our cars, turn on electric lights, or warm last night's pizza in the microwave. However, if a person comes into contact with too much current it can be deadly.

What is Current?

Electric current can be defined as a stream of moving charges, although not all moving charges constitute an electric current. For there to be a current, the net flow of charge passing through a given surface must be greater than zero. In the following diagram, if the net flow of charge is greater than zero, then the light bulb emits light. If the circuit is broken, the current can not flow through the entire circuit and the light bulb does not light up.
currentflow.jpg
Simple Circuit Current Flow

Current is measured in amperes (A) and is denoted by the letter i. It is calculated as follows:
Current.gif
Since current represents the flow of electric charge, we should also be able to associate charge with current. Specifically, we are looking for the charge that passes through an imaginary plane during an interval of time. This is done using the following equation:
Charge.gif
where the time interval is from 0 until t seconds, dt is the differential time interval, and the current i may vary with t.

The SI unit for current is the coulomb (the SI unit of charge) per second (the SI unit of time). So then, one ampere is one coulomb per second: 1 A = 1 C/s.

Current is a scalar, not a vector, because both charge and time are scalars. Current is often represented using arrows to indicate direction of flow, however such arrows are not vectors, but simply tell the reader that the charge is moving.

Charge is conserved, so if a current is split, the sum total of the current afterward is still equal to the original current:
Current_Sum.gif



Example:
Water flows through a garden hose at a volume rate dV/dt of 450 cm^3/s. What is the current of negative charge (the total current)?

The current i of negative charge is due to the electrons in the water molecules moving through the hose. The current is the rate at which that negative charge passes through any plane that cuts completely across the hose.

We can write the current in terms of the number of molecules that pass through such a plane per second as:
Current_3.gif
or as:
Current_3a.gif
We substitute the elementary charge e for "charge per electron", 10 for "electrons per molecule" because a water molecule contains a total of 10 electrons, and dN/dt for "molecules per second". We can express the rate dN/dt in terms of the given volume flow rate dV/dt by first writing:
Mol_Per_Sec.gif
"Molecules per mole" is Avogadro's Number, and "moles per unit mass" is the invers of the mass per mole, which is the molar mass M of the water. "Mass per unit volume" is the (mass) density of water, and "volume per second" is the volume flow rate, dV/dt. Therefore, we have:
dNdt.gif
Substituting this into the equation for i we obtain:
example_current.gif
N is 6.023x10^23 molecules/mol, and density is 1000kg/m^3. By substituting these numbers, along with the molar masses of hydrogen and oxygen, we get:
sample_solution.gif
This current of negative charge is exactly compensated by a current of positive charge associated with the nuclei of the three atoms that make up the water molecule, thus leaving us with the net charge flow of zero.



Current Density

Sometimes we are interested in the current i in a particular conductor, but this is not always the case. Perhaps we would like to look at the flow of charge through a cross section of the conductor at a particular point. To describe this flow, we can use the current density J. We can calculate current in terms of J using the following equation:
current_density.gif
We can also find J in terms of the area A and the current i by using the equation:
current_densitya.gif
Therefore:
current_density_J.gif
And we see that the SI unit for charge density is given as amperes per square meter (A/m^2).

Current density can also be represented using stream lines similar to the electric field lines we studied in a previous chapter. Remember that, since charge is conserved, current density will increase as the area A of the plane that the current is passing through, decreases. Spacing of streamlines will give the observer an idea of the current density at a given point - the closer the streamlines, the greater the current density.

img110.gif
Stream lines representing current density.

Drift Speed

When no current flows through the conductor, the conduction electrons inside it move randomly, with no net motion in any direction. When current does flow through the conductor, the electrons still move randomly, but they also tend to drift with a speed, known unsurprisingly as the drift speed. The drift speed is small compared to the speeds in other directions - it can be many orders of magnitude smaller than the random-motion speed.

When calculating drift speed, we first figure out the total charge q of the charge carries in a given length L with a given cross-sectional area A and a given number of carriers per unit volume n. This is given by:
charge_current_density.gif
where e is the elementary charge. Assuming that all the carriers move along L at the same speed v the total charge moving through any cross-section in a given time interval is
time_interval.gif
We also know that the current i is the time rate of transfer of charge through a cross-section, so now we have:
current_from_charge_time.gif
Using basic algebra to solve for v, we obtain:
drift_speed.gif
Or, in vector form:
drift_speed2.gif
where the quantity ne is the carrier charge density. For positive carriers ne is positive and predicts that J and v have the same direction. For negative carriers, ne is negative and J and v have opposite directions.

Resistance and Resistivity

If we apply the same voltage to different materials, the current through each of them can be very different. This is known as the (electrical) resistance R of the material and is given by:
resistance.gif
The SI unit for resistance is volt per ampere, otherwise known as ohm:
ohm.gif
Given the equation for resistance, it is easy to see that we can get the current for a given resistor:
current_voltage_resistance.gif
Thus, the greater the restistance, the smaller the current.

Resistance is a property of an object; its independent counterpart is (electrical) resistivity, a property of a material. Resistivity is represented by the Greek letter ρ, and calculated as:
resistivity.gif
If we combine the SI units of E and J, we get the ohm-meter as the unit of resistivity:
resistivity_units.gif
This can be represented in vector form as:
resistivity_vector_form.gif
These equations only hold for isotropic materials - materials whose electrical properties are the same regardless of direction.

We can also speak of the (electrical) conductivity of a material, represented by the Greek letter σ. This is simply the reciprocal of resistivity, and is given by:
conductivity.gif
Again, in vector form this is given by:
conductivity_vector_form.gif
The resistivity of certain materials can be found in the table on page 689 of our text book.

Calculating Resistance from Resistivity

If we know the resistivity of a substance (such as copper) we can calculate the resistance of a length of wire made of that substance. If A is the cross-sectional area of the wire, and L is its length, let a potential difference V exist between its ends. We assume that the streamlines representing current density are uniform throughout, and the electric field and current density will be constant. From earlier we have:
calc_res_from_res.gif
We then combine these equations to get:
rho_from_e_and_j.gif
However, since V/i is the resistance R we can now solve for it to obtain:
resistance_from_rho_l_and_a.gif
This equation can only be applied to homogeneous isotropic conductors of uniform cross section.

Variation with Temperature

The properties of most materials change with temperature, and resistivity does as well. It is a nearly-linear relationship that can be written as:
resistivity_from_temp.gif
Here, T0 is the selected reference temperature (usually room temperature, or 293K), and ρ0 is the initial resitivity of the material. The variable α is known as the temperature coefficient of resistivity and is chosen so that the equation gives good agreement with experiment for temperatures in the given range.

Ohm's Law

Ohm's Law can be simply stated as the assertion that the current through a device is always directly proportional to the potential difference applied to the device. This assertion is only true in certain situations, however the term "law" is used for historical reasons.

The law was named after the German physicist Georg Ohm, who, in a treatise published in 1827, described measurements of applied voltage and current through simple electrical circuits containing various lengths of wire. He presented a slightly more complex equation than the one above to explain his experimental results. The above equation is the modern form of Ohm's law.

The resistance of most resistive devices (resistors) is constant over a large range of values of current and voltage. When a resistor is used under these conditions, the resistor is referred to as an ohmic device (or an ohmic resistor) because a single value for the resistance suffices to describe the resistive behavior of the device over the range. When sufficiently high voltages are applied to a resistor, forcing a high current through it, the device is no longer ohmic because its resistance, when measured under such electrically stressed conditions, is different (typically greater) from the value measured under standard conditions.

Ohm's law is an extremely useful equation in the field of electrical/electronic engineering because it describes how voltage, current and resistance are interrelated on a "macroscopic" level, that is, commonly, as circuit elements in an electrical circuit. Physicists who study the electrical properties of matter at the microscopic level use a closely related and more general vector equation, sometimes also referred to as Ohm's law, containing variables that are closely related to the I, V and R scalar variables of Ohm's law, but are each functions of position within the conductor.

Voltage is the electrical driver that moves (negatively charged) electrons through wires and electrical devices, current is the rate of electron flow, and resistance is the property of a resistor (or other device that obeys Ohm's law) that limits current to an amount proportional to the applied voltage. So, for a given resistance R, and a given voltage V established across the resistance, Ohm's law provides the equation I=V/R for calculating the current through the resistor (or device). The following pie chart is useful for finding various characteristics of a circuit including Ohm's law. (Note: Volts (V) has been replaced with (E) in this chart)
ohmspie.jpeg
Ohm's Law Pie Chart

The "device" mentioned by Ohm's law is a circuit element across which the voltage is measured. Resistors are conductors that slow down the passage of electric charge. A resistor with a high value of resistance, say greater than 10 megohms, is a poor conductor, while a resistor with a low value, say less than 0.1 ohm, is a good conductor. (Insulators are materials that, for most practical purposes, do not allow a current when a voltage is applied.)
We can express Ohm's law in a more general way if we focus on conducting materials rather than on conducting devices.

A Microscopic View of Ohm's Law

To understand why Ohm's law is relevant, we need to look at the conduction process at the atomic level. For this we will only look at conducting metals such as copper, for the sake of convenience. If we apply an electric field to a piece of metal, the electrons will change their random motions and drift slowly in the opposite direction of the field, described in the equations for drift speed found earlier. The motion of conduction electrons in an electric field E is a combination of the motion due to random collisions due to E. If an electron mass is put into an electric field of magnitude E, the electron will accelerate according to Newton's 2nd Law:
accel.gif
The nature of these collisions is such that, after colliding, an electron will not "remember" is previous drift velocity. Therefore each collision "resets" the direction and velocity of the electrons involved. The product of the acceleration and the average time between the collisions (represented by the Greek letter τ) is the drift speed of the average electron. So, at any given point in time, the drift speed for an electron will be (on average):
drift_speed_from_accel.gif
Using results found previously, we get:
Elec_from_mass_num_tau_Joules.gif
Which finally leads us to:
rho_from_m_n_and_tau.gif
So we can usually assume that metals will obey Ohm's law if we can show that their resistivity is constant, independent of the strength of the electric field applied.

Power in Electric Circuits

The picture below shows a simple circuit that consists of a battery connected by wires to a light bulb. The battery maintains a potential difference V accros the terminals of the bulb, with a greater potential on the positive (+) side of the bulb than on the negative (-) side.

000688e2.gif
A simple circuit with current flowing clockwise.

The amount of charge dq that moves across the bulb in time interval dt is equal to i*dt. This charge dq moves through a decrease in potential, which means that its electric potential energy decreases by an amount:
dU.gifThe principle of conservation of energy says that the decrease in potential energy is also accompanied by a transfer of energy to some other form. The power P associated with this transfer is given by dU/dt and the equation is:
Power.gif
The SI unit for power is the volt-ampere, which can also be written as one joule per second, also known as the watt. This is seen in the equation below:
watts.gif
For a resistor or some other device with resistance R we can find power if we know the current:
Power_from_i_and_R.gif
and we can also find the power if we know the voltage:
Power_from_V_and_R.gif

Example 1:
Picture_3.png

Example 2:
A common flashlight bulb is rated at 0.30 A and 2.9 V (the values of the current and voltage under operating conditions). If the resistance of the tungsten bulb filament at room temperature (20°C) is 1.1 Ω, what is the temperature of the filament when the bulb is on?
res1.bmp

Example 3:
A student kept his 9.0 V, 7.0 W radio turned on at full volume from 9:00 PM until 2:00 AM. How much charge went through it?

P= i*V
q = it = Pt/V = (7.0 W) (5.0 h) (3600 s/h) / 9.0 Volts = 1.4 X 10^4 C