CK-12 People's Physics Book Version 2
Page 17
In the examples above, was conveniently 90 degrees, which made . But that does not really matter; in a constant magnetic fields a different will simply decrease the force by a constant factor and will not change the qualitative behavior of the particle, since cannot change due to such a magnetic force. (Why? Hint: what is the force perpendicular to? Read the paragraph above.)
Force on a Wire
Since a wire is nothing but a collection of moving charges, the force it will experience in a magnetic field will simply be the vector sum of the forces on the individual charges. If the wire is straight --- that is, all the charges are moving in the same direction --- these forces will all point in the same direction, and so will their sum. Then, the direction of the force can be found using the second right hand rule, while its magnitude will depend on the length of the wire (denoted ), the strength of the current, the strength of the field, and the angle between their directions:
Two current-carrying wires next to each other each generate magnetic fields and therefore exert forces on each other:
By plugging equation [1] into equation [4], one can find the exact formula for this force (left to the reader --- make sure to remember that the two wires can have different currents).
Electromagnetic Induction
Changing magnetic fields passing through a loop of wire generate currents in that wire; this is how electric power generators work. Likewise, a changing current in a wire will create a changing magnetic field; this is how speakers and electric motors work.
To understand induction, we need to introduce the concept of electromagnetic flux. If you have a closed, looped wire of area (measured in ) and loops, and you pass a magnetic field through, the magnetic flux is given by the formula below. Again, the relative direction of the loops and the field matter; this relationship is preserved by creating an 'area vector': a vector whose magnitude is equal to the area of the loop and whose direction is perpendicular to the plane of the loop. The directions' influence can then be conveniently captured through a dot product: The units of magnetic flux are , also known as Webers.
In the example above, there are four loops of wire and each has area (horizontally hashed). The magnetic field is pointing at an angle to the area vector. If the magnetic field has magnitude , the flux through the loops will equal . Think of the magnetic flux as the part of the “bundle” of magnetic field lines “held” by the loop that points along the area vector.
If the magnetic flux through a loop or loops changes, electrons in the wire will feel a force, and this will generate a current. The induced voltage (also called electromotive force, or emf) that they feel is equal to the change in flux divided by the amount of time that change took. This relationship is called Faraday’s Law of Induction:
The direction of the induced current is determined as follows: the current will flow so as to generate a magnetic field that opposes the change in flux. This is called Lenz’s Law. Note that the electromotive force described above is not actually a force, since it is measured in Volts and acts like an induced potential difference. It was originally called that since it caused charged particles to move --- hence electromotive --- and the name stuck (it's somewhat analogous to calling an increase in a particle's gravitational potential energy difference a gravitomotive force). For practical (Ohm's Law, etc) purposes it can be treated like the voltage from a battery.
Since only a changing flux can produce an induced potential difference, one or more of the variables in equation [5] must be changing if the ammeter in the picture above is to register any current. Specifically, the following can all induce a current in the loops of wire:
Changing the direction or magnitude of the magnetic field.
Changing the loops' orientation or area.
Moving the loops out of the region with the magnetic field.
Magnetism Problem Set
Can you set a resting electron into motion with a stationary magnetic field? With an electric field? Explain.
How is electrical energy produced in a dam using a hydroelectric generator? Explain in a short essay involving as many different ideas from physics as you need.
A speaker consists of a diaphragm (a flat plate), which is attached to a magnet. A coil of wire surrounds the magnet. How can an electrical current be transformed into sound? Why is a coil better than a single loop? If you want to make music, what should you do to the current?
For each of the arrangements of velocity and magnetic field below, determine the direction of the force. Assume the moving particle has a positive charge.
Sketch the magnetic field lines for the horseshoe magnet shown here. Then, show the direction in which the two compasses (shown as circles) should point considering their positions. In other words, draw an arrow in the compass that represents North in relation to the compass magnet.
As an electron that is traveling in the positive direction encounters a magnetic field, it begins to turn in the upward direction (positive direction). What is the direction of the magnetic field? direction
direction (towards the top of the page)
direction (i.e. into the page)
direction (i.e. out of the page)
none of the above
A positively charged hydrogen ion turns upward as it enters a magnetic field that points into the page. What direction was the ion going before it entered the field? direction
direction
direction (towards the bottom of the page)
direction (i.e. out of the page)
none of the above
An electron is moving to the east at a speed of . It feels a force in the upward direction with a magnitude of . What is the magnitude and direction of the magnetic field this electron just passed through?
A vertical wire, with a current of going towards the ground, is immersed in a magnetic field of pointing to the right. What is the value and direction of the force on the wire? The length of the wire is .
A futuristic magneto-car uses the interaction between current flowing across the magneto car and magnetic fields to propel itself forward. The device consists of two fixed metal tracks and a freely moving metal car (see illustration above). A magnetic field is pointing downward with respect to the car, and has the strength of . The car is wide and has of current flowing through it. The arrows indicate the direction of the current flow. Find the direction and magnitude of the force on the car.
If the car has a mass of , what is its velocity after , assuming it starts at rest?
If you want double the force for the same magnetic field, how should the current change?
A horizontal wire carries a current of towards the east. A second wire with mass runs parallel to the first, but lies below it. This second wire is held in suspension by the magnetic field of the first wire above it. If each wire has a length of half a meter, what is the magnitude and direction of the current in the lower wire?
Protons with momentum are magnetically steered clockwise in a circular path. The path is in diameter. (This takes place at the Dann International Accelerator Laboratory, to be built in 2057 in San Francisco.) Find the magnitude and direction of the magnetic field acting on the protons.
A bolt of lightening strikes the ground away from a turn coil (see above). If the current in the lightening bolt falls from to in , what is the average voltage, , induced in the coil? What is the direction of the induced current in the coil? (Is it clockwise or counterclockwise?) Assume that the distance to the center of the coil determines the average magnetic induction at the coil’s position. Treat the lightning bolt as a vertical wire with the current flowing toward the ground.
A coil of wire with loops and a radius of is sitting on the lab bench with an electro-magnet facing into the loop. For the purposes of your sketch, assume the magnetic field from the electromagnet is pointing out of the page. In , the magnetic field drops from to . What is the voltage induced in the coil of wire?
Sketch the direction of the current flowing in the loop as the magnetic field is
turned off. (Answer as if you are looking down at the loop).
A wire has of current flowing in the upward direction. What is the value of the magnetic field away from the wire?
Sketch the direction of the magnetic field lines in the picture to the right.
If we turn on a magnetic field of , pointing to the right, what is the value and direction of the force per meter acting on the wire of current?
Instead of turning on a magnetic field, we decide to add a loop of wire (with radius ) with its center from the original wire. If we then increase the current in the straight wire by per second, what is the direction of the induced current flow in the loop of wire?
An electron is accelerated from rest through a potential difference of volts. It then enters a region traveling perpendicular to a magnetic field of . Calculate the velocity of the electron.
Calculate the magnitude of the magnetic force on the electron.
Calculate the radius of the circle of the electron’s path in the region of the magnetic field
A beam of charged particles travel in a straight line through mutually perpendicular electric and magnetic fields. One of the particles has a charge, ; the magnetic field is and the electric field is . Find the velocity of the particle.
Two long thin wires are on the same plane but perpendicular to each other. The wire on the axis carries a current of in the direction. The wire on the axis carries a current of in the direction. Point, has the co-ordinates of in meters. A charged particle moves in a direction of away from the origin at point, , with a velocity of Find the magnitude and direction of the magnetic field at point, .
If there is a magnetic force of on the particle determine its charge.
Determine the magnitude of an electric field that will cancel the magnetic force on the particle.
A rectangular loop of wire long and wide has a resistor of on the side and moves out of a magnetic field at a speed of in the direction of the side. Determine the induced voltage in the loop.
Determine the direction of current.
What would be the net force needed to keep the loop at a steady velocity?
What is the electric field across the long resistor?
What is the power dissipated in the resistor?
A positron (same mass, opposite charge as an electron) is accelerated through volts and enters the center of a long and wide capacitor, which is charged to volts. A magnetic filed is applied to keep the positron in a straight line in the capacitor. The same field is applied to the region (region II) the positron enters after the capacitor. What is the speed of the positron as it enters the capacitor?
Show all forces on the positron.
Prove that the force of gravity can be safely ignored in this problem.
Calculate the magnitude and direction of the magnetic field necessary.
Show the path and calculate the radius of the positron in region II.
f. Now the magnetic field is removed; calculate the acceleration of the positron away from the center. g. Calculate the angle away from the center with which it would enter region II if the magnetic field were to be removed.
A small rectangular loop of wire by moves with a velocity of in a non-uniform field that diminishes in the direction of motion uniformly by . Calculate the induced emf in the loop. What would be the direction of current?
An electron is accelerated through and moves along the positive axis through a plate wide and long. A magnetic field of is applied in the direction. Calculate the velocity with which the electron enters the plate.
Calculate the magnitude and direction of the magnetic force on the electron.
Calculate the acceleration of the electron.
Calculate the deviation in the direction of the electron form the center.
Calculate the electric field necessary to keep the electron on a straight path.
f. Calculate the necessary voltage that must be applied to the plate.
A long straight wire is on the axis and has a current of in the direction. A point , is located above the wire on the axis. What is the magnitude and direction of the magnetic field at .
If an electron moves through in the direction at a speed of what is the magnitude and direction of the force on the electron?
What would be the magnitude and direction of an electric field to be applied at that would counteract the magnetic force on the electron?
Answers to Selected Problems
No: if then ; yes:
.
.
a. Into the page b. Down the page
c. Right
Both pointing away from north
.
.
, south
Down the page;
a. To the right, b.
c. It should be doubled
East
; if CCW motion, B is pointed into the ground.
, counterclockwise
a. b. Counter-clockwise
a. b. Into the page
c.
d. CW
a. b.
c.
E/B
a. b.
a. b. CCW
c.
d.
e.
a. b.
d.
e.
f.
g.
a. b.
c.
d.
e.
f.
a. b.
c.
Chapter 17: Electric Circuits: Advanced Topics
The Big Idea
Modern circuitry depends on much more than just resistors and capacitors. The circuits in your computer, cell phone, and iPod depend on circuit elements called diodes, inductors, transistors, and operational amplifiers, as well as on other chips. In particular the invention of the transistor made the small size of modern devices possible. Transistors and op amps are known as active circuit elements. An active circuit element needs an external source of power to operate. This differentiates them from diodes, capacitors, inductors and resistors, which are passive elements.
Key Concepts
Inductors are made from coiled wires, normally wrapped around ferromagnetic material and operate according to the principles of magnetic induction presented in Magnetism. Inductors generate a back-emf. Back-emf is essentially an induced negative voltage which opposes changes in current. The amount of back-emf generated is proportional to how quickly the current changes. They can be thought of as automatic flow regulators that oppose any change in current. Thus electrical engineers call them chokes.
In a circuit diagram, an inductor looks like a coil. The resistance and capacitance of an inductor are very close to zero. When analyzing a circuit diagram, assume and are precisely zero.
Diodes are passive circuit elements that act like one-way gates. Diodes allow current to flow one way, but not the other. For example, a diode that “turns on” at acts as follows: if the voltage drop across the diode is less than , no current will flow. Above , current flows with essentially no resistance. If the voltage drop is negative (and not extremely large), no current will flow.
Diodes have an arrow showing the direction of the flow.
Transistors are active circuit elements that act like control gates for the flow of current. Although there are many types of transistors, let’s consider just one kind for now. This type of transistor has three electrical leads: the base, the emitter, and the collector.
The voltage applied to the base controls the amount of current which flows from the emitter to the collector.
For example, if the base voltage is more than above the collector voltage, then current can freely flow from the emitter to the collector, as if it were just a wire. If the base voltage is less than above the collector voltage, then current does not flow from the emitter to the collector. Thus the transistor acts as a switch. (This is known as a “diode drop” and varies from transistor to transistor.)
Transistors have an infinite output resistance.. If you measure the resistance between the col
lector and the base (or between the emitter and the base), it will be extremely high. Essentially no current flows into the base from either the collector or the emitter; any current, if it flows, flows from the emitter to the collector..
Transistors are used in amplifier circuits, which take an input voltage and magnify it by a large factor. Amplifiers typically run on the principle of positive and negative feedback. Feedback occurs when a small portion of an output voltage is used to influence the input voltage.
Circuit element Symbol Electrical symbol Unit Everyday device
Voltage Source Volts Batteries, electrical outlets, power stations
Resistor Ohm Light bulbs, toasters, hair dryers
Capacitor Farad Computer keyboards, timers
Inductor Henry Electronic chokes, AC transformers
Diode varies by type none Light-emitting diodes (LEDs)
Transistor varies by type none Computer chips, amplifiers
An operational amplifier or op-amp is an active circuit element that performs a specific function. The most common op-amp has five leads: two input leads, one output lead, and two fixed-voltage leads.
The job of an op-amp is to use the voltage it is supplied to adjust its output voltage. The op-amp will adjust its output voltage until the two input voltages are brought closer together. In other words, the output voltage will change as it needs to until . This won’t happen unless the output voltage is somehow “fed back” into one of the inputs