What is the net charge of the universe? Of your toaster?
As you slide your feet along the carpet, you pick up a net charge of . Which of the following is true? You have an excess of electrons
You have an excess of electrons
You have an excess of protons
You have an excess of protons
You rub a glass rod with a piece of fur. If the rod now has a charge of , how many electrons have been added to the rod?
Not enough information
What is the direction of the electric field if an electron initially at rest begins to move in the North direction as a result of the field? North
East
West
South
Not enough information
Two metal plates have gained excess electrons in differing amounts through the application of rabbit fur. The arrows indicate the direction of the electric field which has resulted. Three electric potential lines, labeled and are shown. Order them from the greatest electric potential to the least.
theyÕre all at the same potential
The diagram to the right shows a negatively charged electron. Order the electric potential lines from greatest to least.
theyÕre all at the same electric potential
The three arrows shown here represent the magnitudes of the electric field and the directions at the tail end of each arrow. Consider the distribution of charges which would lead to this arrangement of electric fields. Which of the following is most likely to be the case here? A positive charge is located at point
A negative charge is located at point
A positive charge is located at point and a negative charge is located at point
A positive charge is located at point and a negative charge is located at point
Both answers a) and b) are possible
Particles and are both positively charged. The arrows shown indicate the direction of the forces acting on them due to an applied electric field (not shown in the picture). For each, draw in the electric field lines that would best match the observed force.
To the right are the electric potential lines for a certain arrangement of charges. Draw the direction of the electric field for all the black dots.
A suspended pith ball possessing of charge is placed away from a metal plate possessing of charge. Are these objects attracted or repulsed?
What is the force on the negatively charged object?
What is the force on the positively charged object?
Calculate the electric field a distance of away from a charge. Then, calculate the force on a charge placed at this point.
Consider the hydrogen atom. Does the electron orbit the proton due to the force of gravity or the electric force? Calculate both forces and compare them. (You may need to look up the properties of the hydrogen atom to complete this problem.)
As a great magic trick, you will float your little sister in the air using the force of opposing electric charges. If your sister has of mass and you wish to float her in the air, how much charge do you need to deposit both on her and on a metal plate directly below her? Assume an equal amount of charge on both the plate and your sister.
Copy the arrangement of charges below. Draw the electric field from the charge in one color and the electric field from the charge in a different color. Be sure to indicate the directions with arrows. Now take the individual electric field vectors, add them together, and draw the resultant vector. This is the electric field created by the two charges together.
A proton traveling to the right moves in between the two large plates. A vertical electric field, pointing downwards with magnitude , is produced by the plates. What is the direction of the force on the proton?
Draw the electric field lines on the diagram.
If the electric field is , what is the acceleration of the proton in the region of the plates?
Pretend the force of gravity doesnÕt exist; then sketch the path of the proton.
We take this whole setup to another planet. If the proton travels straight through the apparatus without deflecting, what is the acceleration of gravity on this planet?
A molecule shown by the square object shown below contains an excess of electrons. What is the direction of the electric field at point A, away?
What is the value of the electric field at point ?
A molecule of charge is placed at point . What are the magnitude and direction of the force acting on this molecule?
Two negatively charged spheres (one with ; the other with ) are apart. Where could you place an electron so that it will be suspended in space between them with zero net force? For problems 19, 20, and 21 assume significant digit accuracy in all numbers and coordinates. All charges are positive.
Find the direction and magnitude of the force on the charge at the origin (see picture). The object at the origin has a charge of , the object at coordinates has a charge of , and the object at coordinates has a charge of . All distance units are in meters.
A charge is located at the origin and a charge is located at . Find the electric field at the coordinate . It may help to draw a sketch.
A metal sphere with a net charge of and a mass of is placed at the origin and held fixed there. Find the electric potential at the coordinate .
If another metal sphere of charge and mass of is placed at the coordinate and left free to move, what will its speed be just before it collides with the metal sphere at the origin?
Collisions of electrons with the surface of your television set give rise to the images you see. How are the electrons accelerated to high speed? Consider the following: two metal plates (The right hand one has small holes allow electrons to pass through to the surface of the screen.), separated by , have a uniform electric field between them of . Find the force on an electron located at a point midway between the plates
Find the voltage difference between the two plates
Find the change in electric potential energy of the electron when it travels from the back plate to the front plate
Find the speed of the electron just before striking the front plate (the screen of your TV)
Two pith balls of equal and like charges are repulsed from each other as shown in the figure below. They both have a mass of and are separated by . One is hanging freely from a string, while the other, also hanging from a string, is stuck like putty to the wall. Draw the free body diagram for the hanging pith ball
Find the distance between the leftmost pith ball and the wall (this will involve working a geometry problem)
Find the tension in the string (Hint: use direction force balance)
Find the amount of charge on the pith balls (Hint: use direction force balance)
Answers to Selected Problems
.
.
.
.
.
.
.
.
.
.
b. c.
a. b.
and . The electric force is orders of magnitudes bigger.
.
a. down b. Up
e.
a. Toward the object b. to the left with a force of
Twice as close to the smaller charge, so from charge and from charge.
and at
and at an angle of from the axis.
a. b.
a. b.
c.
d.
b. c.
d.
Chapter 14: Electric Circuits Version 2
The Big Ideas
In the last chapter, we looked at static configurations of charges. In general, problems with moving charges are very difficult to solve; the field that deals with these is called electrodynamics. In this chapter, we consider how charge can flow through conducting wires connecting opposite ends of a battery. Such a setup, called a circuit usually involves a current, a voltage source, and resistors.
Conductors have an effectively infinite supply of charge, so when they are placed in an
electric field, a separation of charge occurs. A battery with a potential drop across the ends creates such an electric field; when the ends are connected with a wire, charge will flow across it. The term given to the flow of charge is electric current, and it is measured in Amperes (A) --- Coulombs per second. Current is analogous to a river of water, but instead of water flowing, charge does.
Voltage is the electrical energy density (energy divided by charge) and differences in this density (voltage) cause electric current. Batteries often provide a voltage difference across the ends of a circuit, but other voltage sources exist. If current is a river, differences in voltage can be thought of as pipes coming out of a water dam at different heights. The lower the pipe along the dam wall, the larger the water pressure, thus the higher the voltage.
Resistance is the amount a device in the wire resists the flow of current by converting electrical energy into other forms of energy. A resistor could be a light bulb, transferring electrical energy into heat and light or an electric motor that converts electric energy into mechanical energy. The difference in energy density across a resistor or other electrical device is called voltage drop. Resistance is analogous to rocks and other objects that impede the flow of water, transforming the water's kinetic energy into heat, sound, and other forms of energy through contact forces.
This is what a typical circuit looks like:
Circuit Basics
We use the following symbols to represent the quantities discussed above:
Name Symbol Electrical Symbol Units Everyday device
Voltage Volts (V) Battery, the plugs in your house, etc.
Current (flow of charge)
Amps (A)
Whatever you plug into your wall sockets draws current
Resistance Ohm Light bulb, Toaster, etc.
Loop and Junction Rules for Voltage/Current
In electric circuits (closed loops of wire with resistors and constant voltage sources) energy must be conserved. It follows that changes in energy density, the algebraic sum of voltage drops and voltage sources, around any closed loop will equal zero.
In an electric junction or node there is more than one possible path for current to flow. For charge to be conserved at a junction the current into the junction must equal the current out of the junction.
Ohm's Law
The resistance of an object --- described above --- is quantified as the ratio of the voltage drop across it to the amount of current that will flow from that voltage. Note that the current depends on the voltage drop; here, as above we use instead of to mean voltage difference (both are accepted ways). Generally, more current flowing through a resistor will cause a higher voltage drop. For the special class of resistors discussed in this class this ratio is a constant --- the current flowing across these resistors will rise at the same rate as the voltage difference supplied. In other words, the resistance does not depend on the amount of current that flows through the resistor, or the voltage drop across it. This relationship is known as Ohm's Law, for a constant current it is usually written as Unlike equation [1], where varied with current, we can use equation [2] to find the current, voltage drop, or resistance across a resistor when given the other two. When dealing with a constant current, use equation [2], but when dealing with a battery driven circuit (a source of constant voltage difference), use equation [3].
Power
Power is the rate at which energy is lost by a system. The units of power are Watts (W), which equal Joules per second(1W = 1J/s). Therefore, a 60 W light bulb releases 60 Joules of energy every second.
The equation used to calculate the power dissipated in a circuit or across a resistor is: As with OhmÕs Law, one must be careful not to mix apples with oranges. If you want the power of the entire circuit, then you multiply the total voltage of the power source by the total current coming out of the power source. If you want the power dissipated (i.e. released) by a light bulb, then you multiply the voltage drop across the light bulb by the current going through that light bulb.
Resistors in Series and in Parallel
Sometimes, circuits have many resistors in various geometrical arrangements. When in series, two or more resistors are connected end to end (See picture). In this case the resistors receive the same current, but since they can have different resistances they may have different voltage drops across them. Analogously, there may be more rocks at some points in the river than in others, but if there is only one way for the river to flow, the current has to be the same at all points. It follows from Ohm's law that Since the total resistance will increase with each resistor added in series, adding resistors in series will cause the less current to flow at a set voltage (according to Ohm's Law for constant voltage sources, [3]).
When two or more resistors are connected together at both ends, they are said to be "in parallel" (see picture). There are many rivers (the river splits into streams), so all resistors receive different amounts of current. But since they all connect the same points on the circuit, the voltage drops across them have to be equal. The rule for combining resistors in parallel is Since the total resistance will decrease with the number or resistors in parallel, adding resistors in parallel to existing ones will cause more current to flow through a circuit.
Ohm's Law and Total Quantities
Ohm's law is the main relationship for electric circuits but it is often misused. In order to calculate the voltage drop across a light bulb --- or any single resistor --- use the formula: .
Using the formulas and the rules above, a circuit with any number of resistors (and voltage sources) can be modeled as a circuit with just one voltage source and one resistor, for which Ohm's Law also holds. For the total current flowing out of the power source, you need the total resistance of the circuit and the total current: This concept is illustrated below.
Example on Circuit Math
Question: Analyze the diagram below.
a) Find the current going out of the power supply.
b) How many Joules per second of energy is the power supply giving out?
c) Find the current going through the light bulb.
d) Order the light bulbs in terms of brightness.
e) If the light bulbs were all wired in parallel, order them in terms of brightness.
Answer
a) To find the current going out of the power supply, we will use equation [7], . We already have the total voltage drop and we are trying to solve for the current. This means that we need to know the total resistance before we can find the current.
To solve for the resistance we will apply the two rules for resistors (series and parallel) because we have both in are circuit. First, we must combine the two resistors in parallel so that we can treat the entire circuit as a series. According to equation [6], Because is equal to , we need to flip the fraction to get .
Now that we have three resistors in series (the two in parallel can be counted as one), we simply need to add them to get the total resistance.
We can now solve for the current by using equation [7] This is total net current through the circuit; it's also the current across the 50 resistors, but not the ones connected in parallel.
b) To find the power dissipated, we will use equation [4].
c) To find the current going through the light bulb, we must realize that a total of .94A goes through the two light bulbs in parallel; according to the junction rule above, the currents across the two light bulbs must add to this. Now we must find what proportion of the current the light bulb gets. To do this, we use our knowledge that resistors in parallel have the same voltage drop and Ohm's Law: d) The brightness is determined by the power dissipated. More power means a brighter lightbulb. According to equation [4], the power dissipated by a resistor can be written as . Since we know the resistance of and current across every resistor, we can simply calculate this quantity for each one. The order is 's, , then . The is brighter than the because the gets considerably more current.
e) When the bulbs are wired ent
irely in parallel, the voltage drops across them will be the same. Since , the way to determine the brightest bulb is to look at the currents across them, which will be inversely related with their resistances. So, the bulb with the lowest resistance will be the brightest, the one with the second lowest resistance will be second, and so on. Therefore the order is , , and finally .
Capacitors in Circuits (Steady-State)
When a capacitor is placed in a circuit, current does not actually travel across it. Rather, equal and opposite charge begins to build up on opposite sides of the capacitor --- mimicking a current --- until the electric field in the capacitor creates a potential difference across it that balances the voltage drop across any parallel resistors or the voltage source itself (if there are no resistors in parallel with the capacitor). The ratio of charge on a capacitor to potential difference across it is called capacitance:
Capacitors in Series and in Parallel
Like resistors, combinations of capacitors in circuits can be combined into one 'effective' capacitor. The rules for combining them are reversed from resistors:
Charging and Discharging Capacitors (Transient)
When a capacitor is initially uncharged, it is very easy to increase the amount of charge on its plates. As charge builds, the charge present repels new charge with more and more force. Due to this effect, the charging of a capacitor follows a logarithmic curve. When a circuit passes current through a resistor into a capacitor, the capacitor eventually Òfills upÓ and no more current flows across it. A typical RC circuit is shown below; when the switch is closed, the capacitor discharges with an exponentially decreasing current:
CK-12 People's Physics Book Version 2 Page 13
