Charging a capacitor involves moving charges through a potential difference; as we saw in the electricity chapter, this results in electric potential energy being stored in the capacitor:
Capacitor Example
Question: Consider the figure above when switch S is open.
a) What is the voltage drop across the resistor?
b) What current flows through the resistor?
c) What is the voltage drop across the 20 microfarad capacitor?
d) What is the charge on the capacitor?
e} How much energy is stored in that capacitor?
Answer:
a) When the capacitor is charged --- in the steady state --- no current flows across it, and we basically have a circuit with two resistors in series. Accordingly, the voltage drop across the resistor will be in the same proportion to the net voltage across the circuit as its resistance is to the net resistance (see circuits chapter): This means that the voltage drop across the resistor is
b) Since there is only one path for the current to take, its value is the same everywhere on the circuit; all we have to do is find the total current. This will then also be the amount of current that flows through the resistor. We can find it by applying Ohm's Law for the circuit: Since we have the total resistance and the total voltage, we can solve for the total current using Ohm's law. The current flowing through the resistor is therefore .
c) We can find the voltage drop across the capacitor by realizing that the voltage drop across any parallel paths in a circuit have to be equal; otherwise the loop rule would be violated. Therefore, the voltage drop across the capacitor is the same as the voltage drop across the resistor. We can find this analogously to how we found the voltage drop across the other resistor:
d) To find the charge stored in the capacitor we will use the equation First we must convert the capacitor into the correct units for the equation. Then we can substitute in the values and solve for the charge stored.
e) The potential energy stored in a capacitor is All we need to do is plug in the known values and get the potential energy.
Key Terms
DC Power: Voltage and current flow in one direction. Examples are batteries and the power supplies we use in class.
AC Power: Voltage and current flow in alternate directions. In the US they reverse direction 60 times a second. (This is a more efficient way to transport electricity and electrical devices do not care which way it flows as long as current is flowing. Note: your TV and computer screen are actually flickering 60 times a second due to the alternating current that comes out of household plugs. Our eyesight does not work this fast, so we never notice it. However, if you film a TV or computer screen the effect is observable due to the mismatched frame rates of the camera and TV screen.) Electrical current coming out of your plug is an example.
Ammeter: A device that measures electric current. You must break the circuit to measure the current. Ammeters have very low resistance; therefore you must wire them in series.
Voltmeter: A device that measures voltage. In order to measure a voltage difference between two points, place the probes down on the wires for the two points. Do not break the circuit. Volt meters have very high resistance; therefore you must wire them in parallel.
Voltage source: A power source that produces fixed voltage regardless of what is hooked up to it. A battery is a real-life voltage source. A battery can be thought of as a perfect voltage source with a small resistor (called internal resistance) in series. The electric energy density produced by the chemistry of the battery is called emf, but the amount of voltage available from the battery is called terminal voltage. The terminal voltage equals the emf minus the voltage drop across the internal resistance (current of the external circuit times the internal resistance.)
Electric Circuits Problem Set
The current in a wire is 4.5 A. How many coulombs per second are going through the wire?
How many electrons per second are going through the wire?
A light bulb with resistance of is connected to a battery. What is the electric current going through it?
What is the power (i.e. wattage) dissipated in this light bulb with the battery?
How many electrons leave the battery every hour?
How many Joules of energy leave the battery every hour?
A light bulb is shining in your room and you ask yourselfÉ What is the resistance of the light bulb?
How bright would it shine with a battery (i.e. what is its power output)?
A bird is standing on an electric transmission line carrying of current. A wire like this has about of resistance per meter. The birdÕs feet are apart. The bird, itself, has a resistance of about What voltage does the bird feel?
What current goes through the bird?
What is the power dissipated by the bird?
By how many Joules of energy does the bird heat up every hour?
Which light bulb will shine brighter? Which light bulb will shine for a longer amount of time? Draw the schematic diagram for both situations. Note that the objects on the right are batteries, not resistors.
Regarding the circuit to the right. If the ammeter reads , what is the voltage?
How many watts is the power supply supplying?
How many watts are dissipated in each resistor?
Three resistors and one resistor are wired in parallel with a battery. Draw the schematic diagram.
What is the total resistance of the circuit?
What will the ammeter read for the circuit shown to the right?
Draw the schematic of the following circuit.
What does the ammeter read and which resistor is dissipating the most power?
Analyze the circuit below. Find the current going out of the power supply
How many Joules per second of energy is the power supply giving out?
Find the current going through the light bulb.
Find the current going through the light bulbs (hint: it's the same, why?).
Order the light bulbs in terms of brightness
If they were all wired in parallel, order them in terms of brightness.
Find the total current output by the power supply and the power dissipated by the resistor.
You have a power source, two toasters that both run on and a resistor. Show me how you would wire them up so the toasters run properly.
What is the power dissipated by the toasters?
Where would you put the fuses to make sure the toasters don't draw more than 15 Amps?
Where would you put a Amp fuse to prevent a fire (if too much current flows through the wires they will heat up and possibly cause a fire)?
Look at the following scheme of four identical light bulbs connected as shown. Answer the questions below giving a justification for your answer: Which of the four light bulbs is the brightest?
Which light bulbs are the dimmest?
Tell in the following cases which other light bulbs go out if:
bulb goes out (ii). bulb goes out (iii). bulb goes out
Tell in the following cases which other light bulbs get dimmer, and which get brighter if:
bulb goes out (ii). bulb goes out
Refer to the circuit diagram below and answer the following questions. What is the resistance between and ?
What is the resistance between and ?
What is the resistance between and ?
What is the the total equivalent resistance of the circuit?
What is the current leaving the battery?
What is the voltage drop across the resistor?
What is the voltage drop between and ?
What is the voltage drop between and ?
What is the current through the resistor?
What is the total energy dissipated in the if it is in use for 11 hours?
In the circuit shown here, the battery produces an emf of and has an internal resistance of . Find the total resistance of the external circuit.
Find the current drawn from
the battery.
Determine the terminal voltage of the battery
Show the proper connection of an ammeter and a voltmeter that could measure voltage across and current through the resistor. What measurements would these instruments read?
Students measuring an unknown resistor take the following measurements:
Voltage Current
Show a circuit diagram with the connections to the power supply, ammeter and voltmeter.
Graph voltage vs. current; find the best-fit straight line.
Use this line to determine the resistance.
How confident can you be of the results?
Use the graph to determine the current if the voltage were .
Students are now measuring the terminal voltage of a battery hooked up to an external circuit. They change the external circuit four times and develop the following table of data:
Terminal Voltage Current
Graph this data, with the voltage on the vertical axis.
Use the graph to determine the emf of the battery.
Use the graph to determine the internal resistance of the battery.
What voltage would the battery read if it were not hooked up to an external circuit?
Students are using a variable power supply to quickly increase the voltage across a resistor. They measure the current and the time the power supply is on. The following table of data is developed:
Time(sec) Voltage Current
Graph voltage vs. current
Explain the probable cause of the anomalous data after seconds
Determine the likely value of the resistor and explain how you used the data to support this determination.
Graph power vs. time
Determine the total energy dissipation during the seconds.
You are given the following three devices and a power supply of exactly . Device is rated at and Device is rated at and Device is rated at and
Design a circuit that obeys the following rules: you may only use the power supply given, one sample of each device, and an extra, single resistor of any value (you choose). Also, each device must be run at their rated values.
Given three resistors, and and a power source connect them in a way to heat a container of water as rapidly as possible. Show the circuit diagram
How many joules of heat are developed after 5 minutes?
Construct a circuit using the following devices: a power source. Two resistors, device A rated at , ; device rated at , ; device rated at , ; device rated at , .
You have a battery with an emf of and an internal resistance of . Some are drawn from the external circuit. What is the terminal voltage
The external circuit consists of device , and ; device , and , and two resistors. Show how this circuit is connected.
Determine the value of the two resistors.
Students use a variable power supply an ammeter and three voltmeters to measure the voltage drops across three unknown resistors. The power supply is slowly cranked up and the following table of data is developed:
Current Voltage Voltage Voltage
Draw a circuit diagram, showing the ammeter and voltmeter connections.
Graph the above data with voltage on the vertical axis.
Use the slope of the best-fit straight line to determine the values of the three resistors.
Quantitatively discuss the confidence you have in the results
What experimental errors are most likely might have contributed to any inaccuracies.
Design a parallel plate capacitor with a capacitance of . You can select any area, plate separation, and dielectric substance that you wish.
You have a capacitor. How much voltage would you have to apply to charge the capacitor with of charge?
Once you have finished, how much potential energy are you storing here?
If all this energy could be harnessed to lift you up into the air, how high would you be lifted?
Show, by means of a sketch illustrating the charge distribution, that two identical parallel-plate capacitors wired in parallel act exactly the same as a single capacitor with twice the area.
A certain capacitor can store of charge if you apply a voltage of . How many volts would you have to apply to store of charge in the same capacitor?
Why is it harder to store more charge?
A certain capacitor can store of energy (by storing charge) if you apply a voltage of . How many volts would you have to apply to store of energy in the same capacitor? (Important: why isn’t the answer to this just ?)
Marciel, a bicycling physicist, wishes to harvest some of the energy he puts into turning the pedals of his bike and store this energy in a capacitor. Then, when he stops at a stop light, the charge from this capacitor can flow out and run his bicycle headlight. He is able to generate of electric potential, on average, by pedaling (and using magnetic induction). If Mars wants to provide A of current for 60 seconds at a stop light, how big a capacitor should he buy (i.e. how many farads)?
How big a resistor should he pass the current through so the RC time is three minutes?
Given a capacitor with between the plates a field of is established between the plates. What is the voltage across the capacitor?
If the charge on the plates is , what is the capacitance of the capacitor?
If two identical capacitors of this capacitance are connected in series what it the total capacitance?
Consider the capacitor connected in the following circuit at point with two switches and , a resistor and a power source:
i. Calculate the current through and the voltage across the resistor if is open and is closed ii. Repeat if is closed and is open
Figure for Problems 8-10:
Consider the figure above with switch, , initially open: What is the voltage drop across the resistor?
What current flows thru the resistor?
What is the voltage drop across the microfarad capacitor?
What is the charge on the capacitor?
How much energy is stored in that capacitor?
Find the capacitance of capacitors , , and if compared to the capacitor where...
(i). has twice the plate area and half the plate separation (ii). has twice the plate area and the same plate separation (iii). has three times the plate area and half the plate separation
Now the switch in the previous problem is closed. What is the total capacitance of branch II?
What is the total capacitance of branches I, II, and III taken together?
What is the voltage drop across capacitor ?
Reopen the switch in the previous problem and look at the capacitor. It has a plate separation of . What is the magnitude and direction of the electric field?
If an electron is released in the center to traverse the capacitor and given a speed the speed of light parallel to the plates , what is the magnitude of the force on that electron?
What would be its acceleration in the direction perpendicular to its motion?
If the plates are long, how much time would it take to traverse the plate?
What displacement toward the plates would the electron undergo?
With what angle with respect to the direction of motion does the electron leave the plate?
Design a circuit that uses capacitors, switches, voltage sources, and light bulbs that will allow the interior lights of your car to dim slowly once you get out.
Design a circuit that would allow you to determine the capacitance of an unknown capacitor.
The voltage source in the circuit below provides . The resistor is and the capacitor has a value of . What is the voltage across the capacitor after the circuit has been hooked up for a long time?
Answers to Selected Problems
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left = brighter, right = longer
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f. both resistors are brightest, then , then
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.
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.
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CK-12 People's Physics Book Version 2 Page 14
