Physics II: Ch. 21 (Electric Current and Direct-Current Circuits)

Energy Usage

1 kilowatt-hour= 3.6x 10^6 J

Summary of RC circuits

1. Charging and discharging occur over a finite, characteristic time given by the time constant, T=RC 2. At t=o, the current flows freely through a capacitor being charged, it behaves like a short circuit (as if the battery is connected to two resistors R in parallel) 3. As t--> infinity, the current flowing into a capacitor approaches zero. The capacitor now behaves like an open switch because the capacitor plates are fully charged and there is no longer a current.

Big Ideas from the Chapter

1. Electric current is the flow of electric charge from one location to another. By convention, electric current points in the direction that positive charge would flow in a circuit, even though most electric currents are caused by the flow of negative charges. 2. A potential difference is needed to cause electric current to flow through a resistor. The greater the resistance, the greater the required potential difference. 3. Resistors dissipate energy in electric circuits in much the same way that friction acting on a sliding block dissipates mechanical energy. 4. Kirchhoff's rules apply the principle of charge conservation to nodes in a circuit, and the principle of energy conservation to loops in a circuit. 5. In an RC circuit, a finite amount of time is required for charge to build up on a capacitor. Similarly, a finite amount of time is required for a capacitor to discharge. The characteristic time is determined by the product of the resistance (R) and capacitance (C).

Example of Equivalent Capacitance

1. Find the equivalent capacitance of a 10.0 uF capacitor in series with a 5.0 uF capacitor (1/Ceq= 1/10 uF + 1/5uF= 3.33 uF) 2. Find the equivalent capacitance of a 3.33 uF capacitor in parallel with a 20.0 uF capacitor (Ceq= 3.33 uF + 20.0 uF= 23.3 uF) 3. To calculate the energy stored, use the U= 1/2CeqV2 equation

Combination Special

1. First, calculate the Req of the two lower resistors in series. Then this Req=2R so use it as a parallel resistor with the top. Then calculate the Req of the parallel resistors and that's the total Req that can be used to calculate the current. I= emf/Req The total equivalent resistance of the three resistors is less than than the resistance of one of them (this is because more resistors in parallel= smaller equivalent resistance) Also, the current that flows through the lower two resistors is less (because more resistors in series reduces the current (I)). The top resistor would have twice the current (IO than the lower resistors.

Electric circuit

A charge that flows through a closed path and returns to its starting point. The closed path is an electric circuit.

Electric Current

A flow of electric current from one place to another. Carried by electrons moving through a metal wire. (ex: blood cells flowing through an artery). It is the charge that flows past a given point in a wire in a time. 1 Ampere (A)= 1 C/s

Charging a Capacitor

A long time after the switch is closed, the current stops and the capacitor is fully charged. At this point, the voltage across the capacitor is equal to the emf of the battery. Therefore, the charge on the capacitor is Q=Cemf

Graphing Ohm's law

A plot of current vs. voltage for an ohmic material results in a straight line with a constant slope of 1/R.

Current vs. Time for the RC circuit

At t=o, the current is I= emf/R which is the value it would have if the capacitor were replaced by an ideal wire. As t approaches infinity, the current approaches zero because at this point, the capacitor is essentially fully charged so that no more charge can flow onto its plates. It essentially behaves like an open switch now where no more current is flowing.

Using Loops to analyze the circuit

Be sure to use the appropriate signs for currents and potential differences. Can use any two of the three loops It doesn't matter which direction the loop goes in.

Series vs. Parallel Example

Bulbs connected in a series are dimmer than bulbs connected in parallel because in a series, there is less voltage (V) which reduces the power in the bulb (Watts) because P= IV Series = larger equivalent resistance= less current = less power

The limiting behavior of capacitors

Capacitors in RC circuits act like short circuits at t=0 (where all the current is flowing through one wire without a resistor) and open circuits as t approaches infinity (where no more current is flowing because the capacitor has reached its limit on the charge it could hold)

Direct-current circuits (DC circuit)

Circuits with currents that always flow in the same direction.

Alternating-current circuits (AC circuits)

Circuits with currents that periodically reverse their direction.

Capacitors in Series

Combine in the same way as resistors connected in parallel. All three capacitors have charge of the same magnitude on their plates.

Resistor in series Analogy

Connecting the resistors in series is like making a single resistor increasingly longer; as its length increases, so does its resistance.

How RC circuits work

Current flows in the circuit--> leads to a potential drop across the resistor as the current flows through the resistor--> potential difference between the plates of the capacitor will be less than the emf of the battery. This is because every time current flows through a resistor, the electric potential of the charge flowing reduces.

Ammeters

Devices for measuring currents in a circuit An ideal ammeter has zero resistance so it is able to measure the current without altering anything. The ammeter has to be connected in series with the other circuit elements between points A and B.

Capacitors in Parallel

Each capacitor has the same potential difference, emf (V), between its plates The magnitude of the charges on each capacitor are: Q1=C1(emf), Q2=C2(emf), Q3=C3(emf) which is the same as Q=CV Combine in the same way as resistors connected IN SERIES.

Speed of Conduction Electrons

Electrons constantly collide with the atoms in the wire as they are flowing so their speed is rather slow. Drift speed= electron's average speed (limited by repeated collisions) As an electron moves away from the battery, it exerts a force on its neighbors which causes them to move in the same direction and this domino effect is what causes electrons to move at the speed of light.

Electron flow in a battery

Electrons do not flow in metal wires unless the wires are connected to a source of electrical energy (which has high electric potential).

Ohm's Law

Electrons experience resistance to their motion as they flow through a wire (just like a box sliding across the floor experiences friction) To move electrons against the resistance of a wire, MUST apply a potential difference between the ends (similar to the water flow in a garden hose) That potential difference that must be applied to create a current even with resistance is given by Ohm's law. Can solve for the resistance an electron is experiencing in a wire using Ohm's law. 1 ohm= 1 V/A The greater the resistance, the greater the required potential difference.

Finding the time in the charge vs. time equation

Ex: to find the time when the charge will be 80% of the full charge, set q(t)= 0.800Cemf 0.800Cemf=Cemf (1-e^-t/T) And we know that T=RC

Equivalent Capacitance for Capacitors in Parallel

If an equivalent capacitor is used to replace the three in parallel, the charge on its plates must be the same as the total charge on the individual capacitors: Q=(Ceq)(emf) Connecting capacitors in parallel produces an equivalent capacitance greater than the greatest individual capacitance.

Short circuit

If any of the resistors in a parallel connection is equal to zero, the equivalent resistance is also zero. All of the current flows through the path of zero resistance so 1/R2 and 1/R3= 0 and if 1/R1=0, then 1/Req must also equal zero.

Current in an RC Circuit

Immediately after the switch is closed, the capacitor acts like a short circuit so as if the resistors are in parallel and the current, I=emf/(R/2)= 2emf/R After the current has been flowing for a while, the capacitor acts like an open switch. Now the current can only flow through the one resistor without the capacitor so the current is now I=emf/R; half of its initial value.

Discharging a capacitor

Initially, the circuit is open so no current can flow. When the switch is closed, current flows from the high potential (+) plate to the low potential (-) plate. Eventually, the charge remaining on the capacitor approaches zero as time goes by.

Voltmeter

Measures the potential drop between any two points in a circuit. Must be connected "in parallel" to the circuit between points C and D.

Electrical Power

Power is the rate at which energy changes. (delta U/delta t). A current flowing through a resistor dissipates power (energy per time). Therefore, the smaller the power (means a lot of energy has been dissipated by the resistor), the greater the resistance. Power= Watts (W) In a resistor, power is dissipated in the form of heat.

Internal Resistance (r)

Real batteries have internal losses that cause the potential difference between their terminals to be less than the emf If a battery is connected to an external resistance R, the equivalent resistance of the circuit is r + R The potential difference between the terminals of the battery is emf- Ir which is smaller than the emf of the battery. Smaller potential difference= smaller current

Resistors in Parallel

Resistors that are connected across the same potential difference The current has parallel paths through which it can flow Total current is equal to the sum of the currents through EACH of the three resistors Potential difference is the same for each of the resistors

Resistors in Series

Resistors that are connected one after the other are in series. The three resistors acting together have the same effect, they draw the same current as a single equivalent resistor (Req). The same current (I) must flow through each of the resistors. The total electric potential in the current is the sum of the electric potentials at each resistor (V1= IR1, V2= IR2, V3=IR3, etc)= so the total potential difference from point A to point B must be the emf of the battery.

V= Req

Same equation as:

Direction of current flow

The direction of the current in an electric circuit is the direction in which a positive test charge would move. A positive test charge would move in the direction of high to low electric potential. However, negatively charged electrons are flowing from the negative terminal of the battery to the positive terminal. The direction of the current (I) is opposite the direction of the flow of electrons.

Electromotive force (emf)

The emf of a battery is the potential difference it can produce between its terminals when it is disconnected from a circuit and carries no current. The emf determines the amount of work a battery does to move a certain amount of charge around a circuit. Unit of emf= Volts (V) The more charge a battery moves through a circuit, the more work it does and the greater the emf, the greater the work done.

Equivalent Capacitance for Capacitors in Series

The equivalent capacitance of a group of capacitors connected in series is less than the smallest individual capacitance (just like resistors in parallel). The plate separations of the individual capacitors add to give a larger effective separation= smaller capacitance. (C is inversely proportional to d)

Equivalent Resistance for Resistors in Series

The equivalent resistance is simply the sum of the individual resistances. (Req= R1 + R2 + R3) The equivalent resistance is greater than the greatest resistance of any of the individual resistors.

General rule of equivalent resistors in parallel

The equivalent resistance of resistors in parallel is less than or equal to the resistance of the smallest resistor. (This is because more resistors are more paths that the current could flow through which means a smaller equivalent resistance) Parallel circuits tend to have more current than resistors in series because their equivalent resistances are small.

Relationships between power, V, and R

The greater the power, the greater the current (I). Or...the greater the change in electrical energy over time, the greater the current. To get the amount of energy dissipated, U, multiply power by change in time by calculating the power either from the battery emf (V) and R or from the current (I) and R, etc.

Charge vs. Time for the RC Circuit

The limit as t goes to infinity is the charge (Q) the capacitor would have had from the beginning if there had been no resistor in the circuit.

Equivalent Resistance for Resistor in Parallel

The more resistors are connected in parallel, the smaller the equivalent resistance. This is because there's more paths through which the current can flow (equivalent to using a wire with a greater area). More current flowing with the same potential difference means that the equivalent resistance has been reduced. (R= V/I) After summing the inverses of resistors in parallel, remember to take one more inverse at the end of your calculation to find the equivalent resistance

Potential difference for capacitors in series

The potential difference across the three capacitors must equal the emf of the battery.

Resistivity

The resistance of a wire depends on the particular material from which it is constructed and on its length and area. The quantity that characterizes the resistance of a given material is its resistivity. The greater the resistivity of a material, the greater the resistance. The larger the area of the wire, the less resistance the electrons will experience and the longer the wire, the more resistance the electron will experience. The area of a wire is piD^2/4 Resistivity increases as temperature increases.

Kirchhoff's Junction Rule

The sum of all currents meeting at any junction in a circuit must equal zero. The current entering any point in a circuit must equal the current leaving that point (otherwise charge would either build up or disappear from a circuit) A junction is any point in a circuit where three or more wires meet.

Kirchhoff's Loop Rule

The sum of all potential differences around any closed loop in a circuit is zero. From point A to B, the electric potential (emf) increases. From point B to C, it is an ideal wire so there is no potential change. From point C to D, the potential changes because a potential difference is required to force a current through a resistor. From point D to A, there is no potential difference because it's an ideal wire. deltaVcd= -(emf)

Internal Resistance in a Battery

There is always some internal loss of potential in a real battery. The greater the current flowing through a battery, the greater the reduction in potential difference between its terminals. Only when the current is zero can a real battery produce its full emf.

Energy in Parallel capacitors

To find the capacitance of one the capacitors in parallel, use the U=1/2 CV equation to solve for Ceq then use the fact that Ceq= C1 + C2 to solve for any of their capacitance. The larger capacitor (the one with the greater capacitance) stores the greater amount of energy (U)

Power in Series and Parallel

Total power dissipated in a parallel circuit is much greater than that dissipated in the series circuit. This is because the equivalent resistance of the parallel circuit is smaller than the equivalent resistance of the series circuit (P=V2/R) Smallest resistor= largest current (I)= largest power in the parallel circuit

Battery

Uses chemical reactions to produce a difference in electric potential between its two terminals. The terminal with high electric potential is + and the terminal with low electric potential is - When the battery is connected to a circuit, electrons move in a closed path from the negative terminal of the battery, through the circuit and back to the positive terminal. In an open circuit, there is no closed path through which the electrons can flow so the light cannot go on.

Combination Circuits (Series + Parallel)

When considering an electric circuit with resistors in series and parallel, work from the smallest units of the circuit outward to even larger units. (smallest units are the parallel resistors)

Simple RC Circuits

Without a resistor, when the switch is closed on a circuit with only batteries and capacitors, the charge on the capacitor plates would appear immediately. However, resistors limit the rate at which charge can flow and slow the charging process of the capacitor plates. The larger the resistance, the longer it takes for the capacitor to charge.