Current 1. Charges in motion 1. Cause of motion 2. Where is it going? 3. Let's keep this going. 2. Current 1. Flow of particles in pipes. 2. A constant problem 3. Conservation Laws 4. Microscopic motion of Charges in conductors 1. Drift Velocity 2. Current density 3. Batteries 5. Circuits: Basics 1. Ohmic vs. Non-Ohmic 2. Resistance vs. Resistivity 3. Conductivity vs. Conductance 1. Charges in motion source charges test charg Fig. 1 Electrostatics - sources are stationary, test charges might might move. So far, we ve looked at charges which are essentially stationary. We ve assumed that metals are in equilibrium, and if a test charge does move, that s the only thing moving. Now, we'll start talking about moving charges. We need to consider two aspects: 1. A reason to move. 2. A place to move to. updated on 2018-04-12 J. Hedberg 2018 Page 1
Cause of motion Fig. 2 We start with two neutral conductors. Through some process, we charge one positively, and one negatively. Since like charges are near each other, there will be repulsive forces trying to spread them out. But, there s no where to go... updated on 2018-04-12 J. Hedberg 2018 Page 2
1.5 V Where is it going? Now, we connect the two charged conductors with another conductor. This conducting path provides a place for the charges to go. They will move through the conducting path until everything is neutral. Let's keep this going. At this point, when the two sources are neutral, there will be no more flow. So, we need something which will continually separate the charges. That'll be a battery. updated on 2018-04-12 J. Hedberg 2018 Page 3
If we look closely at our conducting path, we will find that it s actually only the negative charges that are moving. (The electrons) There are stationary positive nuclei in the conductor. The negative electrons however, are free to move around. Fig. 3 A microscopic view of the current carrying wire. Quick Question 1 Which one of the following statements concerning the electric field inside a conductor is true if electrons are moving from right to left in a conducting wire? 1. The electric field must be zero in this case. 2. The electric field is directed perpendicular to the direction the electrons are moving. 3. The electric field is directed toward the left. 4. The electric field is directed toward the right. updated on 2018-04-12 J. Hedberg 2018 Page 4
The electron does not move straight through the conductor, but instead undergoes a series of collisions with the stationary nuclei. However, the electric field keep the charge moving on average in a certain direction. Fig. 4 We ll see that the average velocity of the moving charge is quite slow. It s called the drift velocity. 2. Current Let's define electrical current: I = dq dt In SI, it has the fundamental unit called an ampere, which is defined as: 1 A = 1 C 1 s Flow of particles in pipes. The pipe on the left has a constriction. This prevents reduces the number of particles passing through a point at a given time. updated on 2018-04-12 J. Hedberg 2018 Page 5
(+) postive (Higher V) I Current: direction a positive charge would go (even thought it's actually negative charges moving) ( ) negative (Lower V) Fig. 5 Since the battery keeps separating the charges inside it, this creates one electrode with a higher electric potential, and one with a lower. This potential difference will cause charges to move in the connecting wire. Even though it s electrons which are moving in real metals, the current is defined as the direction a positive charge would go. A constant problem There are two ways of talking about the direction of charge flow. electron current electric field & conventional current The electrons are the actual charge carriers in metal, but we say the current is traveling in the direction a positive charge would go. (which is also the direction of the electric field) (It's a known issue) updated on 2018-04-12 J. Hedberg 2018 Page 6
3. Conservation Laws (+) postive (Higher V) I A first principle about electric currents is the conservation of current: The current is the same at all points in a current-carrying wire. This means that the rate of flow of charges through a given point is the same for every point along the wire. ( ) negative (Lower V) (+) postive (Higher V) ( ) negative (Lower V) Fig. 6 I A B The law of conservation of current tells us that at points, A and B in this diagram, the number of charges passing each second is the same. The light bulb DOES NOT use up charges. All the charges that were flowing before, at point A, are still there at point B. Charge flowing through a wire or light bulb is just like this water mill. The amount of water coming into the top of the wheel equals the amount coming out at the bottom. Flow is conserved. updated on 2018-04-12 J. Hedberg 2018 Page 7
In the water analogy, the sum of the water entering a fork in the river equals the sum of the water that flows out. Fig. 7 The same is true for electrical currents. I in = I out Kirchoff s Junction Law:The sum of currents into a junction equals the sum out of that junction. updated on 2018-04-12 J. Hedberg 2018 Page 8
Quick Question 2 Here is a network of wire meeting at a junction (node). What is the flow in the branch with the question mark?.3 A.2 A.5 A.4 A? 1..3 A into the node 2..4 A out of the node 3..4 A into the node 4. 0.0 A in that branch 4. Microscopic motion of Charges in conductors An electron will scatter off of the ions in the metal. If there is no average electric field in the wire, the process will lead to no net motion of the electron. Fig. 8 updated on 2018-04-12 J. Hedberg 2018 Page 9
However, if there is an electric field present in the wire, then the electron will move in parabolic arcs, due to the constant acceleration in one direction. Eventually, there will be a net motion in one direction. Fig. 9 v Or, we can think about the velocity verses time graphs for these moving objects. The top graph shows one collision. t After many collisions, since the acceleration is always in the same direction, due to the electric field, we would expect a non zero average velocity. This is the 'drift velocity'. v We can also point out the time between each collision, or scattering event, can be called Δt. v(avg) t Fig. 10 updated on 2018-04-12 J. Hedberg 2018 Page 10
Since F = ma, and ee = F, we can consider the acceleration in the horizontal direction to be given by: a x = ee m This acceleration in the x direction will cause the electron's velocity to increase with time: ee v x = v 0 + a x Δt = v 0 + Δt m And so, the average initial velocty of the electrons will be zero, and the average time between collisions is τ, we can say the average velocity of all the electrons in the wire will be given by: ee v x = v 0 + Δt m v d = ee τ m Drift Velocity The drift velocity is calculated using a few parameters from the wire. Let s say there are n charges per unit volume in a conductor. Then the number of charges in a length Δx will be: N = ΔxAn Multiplying this by the charge per carrier, q, gives: The drift speed, v d, is the rate at which the charge carriers move: = So, ΔQ = (naδx)q ΔQ = naq v d Δt. v d Δx Δt And, current is ΔQ Δt, thus: I = nq v d A updated on 2018-04-12 J. Hedberg 2018 Page 11
Combining the two formulas for the drift velocity: v d = eeτ m and I = nq allows us to get another, more microscopically informed function for the current in a wire. v d A I = n e e 2 E mτa A n e m E τ e is the cross-sectional area is the electron density is the electron mass is the electric fields strength is the mean time between collisions is the charge on the electron Example Problem #1: Estimate how long it will take for an electron to go from a battery to a light bulb 10 meters away. Quick Question 3 If I double the diameter of a conducting wire, how might this effect the drift velocity of the charges? The drift velocity: 1....will be doubled 2....will be half as slow 3....will be four times as fast 4....will be four times slower 5....will not change. updated on 2018-04-12 J. Hedberg 2018 Page 12
Current density We can also talk about the current per cross sectional area, the current density. Since the current everywhere in this diagram of a wire (river) has to be the same, the density will change due to changes in width. The current density is written: J = I/A We talked qualitatively about some properties of conduction paths which affect the currents in those paths. 1.5 V Things in here affect the current! The collisions between electrons and nuclei, the wire diameter, temperature, etc. Let s go through piece by piece and see how I is changed. updated on 2018-04-12 J. Hedberg 2018 Page 13
Batteries We need a way to separate charges besides rubbing rods and wool or using giant metal spheres: We won't worry about what's going on inside the battery right now, but we can just consider it to be a little black box that separates charge from one side to another. 4.5 V Before we get to things inside the box, let s think about the potential difference of the battery first. 3 V Ohm figured out that the magnitude of the current in a wire was proportional to the potential difference between it s ends. I V 1.5 V I 1 I 2 I 3 0 V 0 V 0 V Georg Simon Ohm [1789-1854] updated on 2018-04-12 J. Hedberg 2018 Page 14
Whenever we have a proportionality, we can make it an equality by using a constant: V = RI This R will be our constant of proportionality between current and voltage. (Of course, we can call it resistance. ) Resistance tells us exactly what it should: how hard is it to pass charges through a given path. Its SI unit the Ohm: Ω This relation is known as Ohm's law. 5. Circuits: Basics V = IR In general, our goal will be to assess the ciruit in order to get two of these terms, then use them to find the third. I 1.5 V Things in here affect the current! + V R updated on 2018-04-12 J. Hedberg 2018 Page 15
Ohmic vs. Non-Ohmic resistor wire wire The normal resistor is an example of an Ohmic device. ohmic non-ohmic I I V V Ohmic devices will have a linear relationship between current and voltage. The resistance is a constant for Ohmic devices. (It could change based on applied voltage or current in non-ohmic devices) Resistance vs. Resistivity resistivity: is a property of materials. It captures all the physics of the internal collisions and other causes of charge impediments so we can calculate the objects resistance. For copper, it s equal to 1.7 10 8 Ωm A L R = ρ L A Resistance is the property of a particular device. updated on 2018-04-12 J. Hedberg 2018 Page 16
Quick Question 4 Which question(s) is(are) nonsense? 1. What is the resistivity of Aluminum? 2. What is the resistance of pure gold? 3. What is the resistance of that piece of copper? 4. Which of these two silver wires has a larger resistivity? 5. Where should I cut this wire to make its resistivity half as much? Some resistivity values of common materials. Materal resistivity ( Ωm) Silver 1.59 10 8 Copper 1.68 10 8 Gold 2.44 10 8 Iron 9.7 10 8 Salty Sea Water.22 Tap Water 20 200 Glass 10 10 10 10 14 Teflon 10 10 22 These values can be temperature dependeant, but otherwise are essentially determined by the elemental makeup of the material. We can also define the resistivity of a material as the ratio of the electric field at a given point divided by the current density. ρ = E J or, in vectors: E = ρj updated on 2018-04-12 J. Hedberg 2018 Page 17
Quick Question 5 A copper wire is made in the shape shown, which has a gradually decreasing diameter along its length. If an electric current is moving through the wire, which quantities vary along the length of the wire? 1. current only 2. current and current density only 3. current density and electric field only 4. resistivity and current only 5. current, resistivity, current density, and electric field Conductivity vs. Conductance Likewise, we can define a conductivity which will just be the reciprocal of resistivity: σ = 1 ρ The inverse of resistance will be called conductance: G 1 G = = R I V updated on 2018-04-12 J. Hedberg 2018 Page 18
Quick Question 6 Over time the power cable between these two poles sags due to gravity. Assuming the voltage provided by the power station remains the same, will the current through the wire: 1. Increase 2. stay the same 3. decrease updated on 2018-04-12 J. Hedberg 2018 Page 19