Showing posts with label ELECTROMAGNETICS. Show all posts
Showing posts with label ELECTROMAGNETICS. Show all posts

Magnetic force on a current carrying wire virtual experiment

Quiz


Q1) The force on a moving charge in the presence of a magnetic field is always

A. Perpendicular to both the charge's velocity and the magnetic field
B. Parallel to the charge's velocity
C. Parallel to the magnetic field
D. None of the above

Answer) A.

Q2)  A current is moving from north to south through a long wire that is lying horizontally on a table. What is the direction of the magnetic force if the magnetic field is directed up and out of the table?

A. toward the north
B. toward the east
C. toward the south
D. toward the west

 Answer) D.

Magnetic force on a current carrying wire Simulation(Virtual Experiment)


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Magnetic force on a current carrying wire
If a magnetic field exerts a force on a single charged particle when it moves through a magnetic field, it should be no surprise that magnetic forces are exerted on a current-carrying wire as well (see Fig. 1). Because the current is a collection of many charged particles in motion, the resultant force on the wire is due to the sum of the individual forces on the charged particles. The force on the particles is transmitted to the “bulk” of the wire through collisions with the atoms making up the wire.

FIGURE 1 This apparatus demonstrates the force on a current carrying wire in an external magnetic field. Why does the bar swing away from the magnet after the switch is closed?


Some explanation is in order concerning notation in many of the figures. To indicate the direction of $\ B$, we use the following conventions:


If $\ B$ is directed into the page, as in Figure 2, we use a series of green crosses, representing the tails of arrows. If $\ B$ is directed out of the page, we use a series of green dots, representing the tips of arrows. If $\ B$ lies in the plane of the page, we use a series of green field lines with arrowheads.

The force on a current-carrying conductor can be demonstrated by hanging a wire between the poles of a magnet, as in Figure 2. In this figure, the magnetic field is directed into the page and covers the region within the shaded area. The wire deflects to the right or left when it carries a current.

FIGURE 2  A segment of a flexible vertical wire partially stretched between the poles of a magnet, with the field (green crosses) directed into the page. (a) When there is no current in the wire, it remains vertical. (b) When the current is upward, the wire deflects to the left. (c) When the current is downward, the wire deflects to the right.



Magnetic Force on a wire resulting from a magnetic field 
It is possible to determine the force of magnetism exerted on a current-carrying wire passing through a magnetic field at right angles to the wire. Experiments show that the magnitude of the force, $\ F$, on the wire, is proportional to the strength of the field, $\ B$, the current, $\ I$, in the wire, and the length, $\ L$, of the wire in the magnetic field. The relationship of these four factors is as follows:



Magnetic Force on a Current-Carrying Wire in a Magnetic Field 

$\ F=BIL$

The force on a current-carrying wire in a magnetic field is equal to the product of magnetic field strength, the current, and the length of the wire.

The strength of a magnetic field, $\ B$, is measured in teslas, T. 1T is equivalent to 1 N/A m.
Note that if the wire is not perpendicular to the magnetic field, a factor of $\sin \theta$ is introduced in the above equation, resulting in $\ F=BILsin \theta$ . As the wire becomes parallel to the magnetic field, the angle becomes zero, and the force is reduced to zero. When $\theta$=90°, the equation is again $\ F=BIL$.



Determining the Magnetic force’s direction 
Faraday’s description of the force on a current-carrying wire does not completely describe the direction because the force can be upward or downward. The direction of the force on a current carrying wire in a magnetic field can be found by using the third right-hand rule. This technique is illustrated in Figure 3. The magnetic field is represented by the symbol B, and its direction is represented by a series of arrows. To use the third right-hand rule, point the fingers of your right hand in the direction of the magnetic field, and point your thumb in the direction of the conventional (positive) current in the wire.

Figure 3 The third right-hand rule can be used to determine the direction of force when the current and magnetic field are known.

The palm of your hand will be facing in the direction of the force acting on the wire. When drawing a directional arrow that is into or out of the page, direction is indicated with crosses and dots, respectively. Think of the crosses as the tail feathers of the arrow, and the dots as the arrowhead.

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Superposition principle of electric field virtual experiment

Quiz


Q1) A has a charge of +2 µC, and object B has a charge of +6 µC. Which statement is true:

A. $\ F_{AB} $= $\ –3F_{BA}$
B. $\ F_{AB} $=$\ –F_{BA}$
C. $\ 3F_{AB} $=$\ –F_{BA}$

Answer) B.
By Newton’s third law, the two objects will exert forces having equal magnitudes but opposite directions on each other.

Q2) At the position of the dot, the electric field points

A. Up.
B. Down.
C. Left.
D. Right.
E. The electric field is zero.

 Answer) A.
The direction of the field is taken to be the direction of the force it would exert on a positive test charge

Superposition principle of electric fields Simulation(Virtual Experiment)


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Force, Charges, and Distance 
The electrical force between any two objects obeys a similar inverse-square relationship with distance. The relationship among electrical force, charges, and distance, now known as Coulomb’s law, was discovered by the French physicist Charles Coulomb (1736–1806) in the eighteenth century. Coulomb’s law states that for charged particles or objects that are small compared with the distance between them, the force between the charges varies directly as the product of the charges and inversely as the square of the distance between them.

Coulomb’s law can be expressed as

$\ F = k \frac {q_{A}q_{B}}{d^2}$

where $\ d$ is the distance between the charged particles; $\ q_{A}$ represents the quantity of charge of one particle and $\ q_{B}$the quantity of charge of the other particle; and $\ k$ is the proportionality constant.


When the charges are measured in coulombs, the distance is measured in meters, and the force is measured in newtons, the constant, K, is $\ 9.0 X 10^9  N m^2/C^2$.
This equation gives the magnitude of the force that charge $\ q_{A}$ exerts on $\ q_{B}$ and also the force that $\ q_{B}$ exerts on $\ q_{A}$. These two forces are equal in magnitude but opposite in direction. You can observe this example of Newton’s third law of motion in action when you bring two strips of tape with like charges together. Each exerts forces on the other. If you bring a charged comb near either strip of tape, the strip, with its small mass, moves readily. The acceleration of the comb and you is, of course, much less because of the much greater mass.
The electrical force, like all other forces, is a vector quantity. Force vectors need both a magnitude and a direction. However, Coulomb’s law will furnish only the magnitude of the force. To determine the direction, you need to draw a diagram and interpret charge relations carefully.
Consider the direction of force on a positively charged object called A. If another positively charged object, B, is brought near, the force on A is repulsive. The force, $\ F_{B}$ on A, acts in the direction from B to A, as shown in Figure 1a. If, instead, B is negatively charged, the force on A is attractive and acts in the direction from A to B, as
shown in Figure 1b.


FIGURE 1 The rule for determining direction of force is like charges repel, unlike charges attract.

The superposition principle
• The electric force obeys the superposition principle.
• That means the force two charges exert on a third force is just the vector sum of the forces from the two charges, each treated without regard to the other charge.
• The superposition principle makes it mathematically straightforward to calculate the electric forces exerted by distributions of electric charge.
• The net electric force is the sum of the individual forces.





The Electric Field
The electric field is a vector quantity that relates the force exerted on a test charge to the size of the test charge. How does this work? An electric charge, q, produces an electric field that you can measure. This is shown in Figure 2. First, measure the field at a specific location.

FIGURE 2 Arrows can be used to represent the magnitude and direction of the electric field about an electric charge at various locations.

Call this point A. An electrical field can be observed only because it produces forces on other charges, so you must place a small positive test charge, $\ q′$, at A. Then, measure the force exerted on the test charge, $\ q′$, at this location.
According to Coulomb’s law, the force is proportional to the test charge. If the size of the test charge is doubled, the force is doubled.
Thus, the ratio of force to test charge is independent of the size of the test charge. If you divide the force, F, on the test charge, measured at point A, by the size of the test charge, $\ q′$, you obtain a vector quantity, $\ F/q$′. This quantity does not depend on the test charge, only on the charge q and the location of point A. The electric field at point A, the location of $\ q′$, is represented by the following equation.

Electric Field E

$\ E =  \frac {F_{on q'}}{q'}$
$\ E =  k\frac {q}{r^2}$.....[single point charge q]

The direction of the electric field is the direction of the force on the positive test charge. The magnitude of the electric field is measured in newtons per coulomb, N/C.
A picture of an electric field can be made by using arrows to represent the field vectors at various locations, as shown in Figure 2. 

The length of the arrow will be used to show the strength of the field. The direction of the arrow shows the field direction. To find the field from two charges, the fields from the individual charges are added vectorially. A test charge can be used to map out the field resulting from any collection of charges. 


Superposition principle of electric fields
The superposition principle states that, for a collection of charges, the total electric field at a point is simply the summation of the individual fields due to each charge


at any point P, the total electric field due to a group of charges equals the vector sum of the electric fields of the individual charges.




The dipole: an important charge distribution 
• An electric dipole consists of two point charges of equal magnitude but opposite signs, held a short distance apart.

Faraday’s Law and Lenz's Law Electromagnetic Induction Simulation

Quiz


Q1) Faraday’s law says that

A. an emf is induced in a loop when it moves through an electric field
B. the induced emf produces a current whose magnetic field opposes the original change
C. the induced emf is proportional to the rate of change of magnetic flux

Answer) C.


Q2)  If a coil is shrinking in a B field pointing into the page, in what direction is the induced current?

A. clockwise
B. counter-clockwise
C. no induced current

 Answer) A.
Downward flux is decreasing, so need to create downward B field.

Faraday’s Law Electromagnetic Induction Simulation(Virtual Experiment)


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Faraday and Henry both made the same discovery. Electric current can be produced in a wire by simply moving a magnet into or out of a wire coil. No battery or other voltage source was needed only the motion of a magnet in a coil or in a single wire loop as shown in Figure 1.

 FIGURE 1. When the magnet is plunged into the coil, voltage is induced in the coil and charges in the coil are set in motion.

They discovered that voltage was induced by the relative motion of a wire with respect to a magnetic field. The production of voltage depends only on the relative motion of the conductor with respect to the magnetic field. Voltage is induced whether the magnetic field of a magnet moves past a stationary conductor, or the conductor moves through a stationary magnetic field as shown in Figure 2. The results are the same for the same relative motion.

FIGURE 2. Voltage is induced in the wire loop whether the magnetic field moves past the wire or the wire moves through the magnetic field.


Faraday’s Law of Electromagnetic Induction
To see how an emf can be induced by a changing magnetic field, consider a loop of wire connected to a sensitive ammeter, as illustrated in Figure 3. When a magnet is moved toward the loop, the galvanometer needle deflects in one direction, arbitrarily shown to the right in Figure 3a. When the magnet is brought to rest and held stationary relative to the loop (Fig. 3b), no deflection is observed. When the magnet is moved away from the loop, the needle deflects in the opposite direction, as shown in Figure 3c. Finally, if the magnet is held stationary and the loop is moved either toward or away from it, the needle deflects. From these observations, we conclude that the loop detects that the magnet is moving relative to it and we relate this detection to a change in magnetic field. Thus, it seems that a relationship exists between current and changing magnetic field.
These results are quite remarkable in view of the fact that a current is set up even though no batteries are present in the circuit! We call such a current an induced current and say that it is produced by an induced emf.

Figure 3 (a) When a magnet is moved toward a loop of wire connected to a sensitive ammeter, the ammeter deflects as shown, indicating that a current is induced in the loop. (b) When the magnet is held stationary, there is no induced current in the loop, even when the magnet is inside the loop. (c) When the magnet is moved away from the loop, the ammeter deflects in the opposite direction, indicating that the induced current is opposite that shown in part (a). Changing the direction of the magnet’s motion changes the direction of the current induced by that motion.


Now let us describe an experiment conducted by Faraday and illustrated in Figure 4. A primary coil is connected to a switch and a battery. The coil is wrapped around an iron ring, and a current in the coil produces a magnetic field when the switch is closed. A secondary coil also is wrapped around the ring and is connected to a sensitive ammeter. No battery is present in the secondary circuit, and the secondary coil is not electrically connected to the primary coil. Any current detected in the secondary circuit must be induced by some external agent.

Initially, you might guess that no current is ever detected in the secondary circuit. However, something quite amazing happens when the switch in the primary circuit is either opened or thrown closed. At the instant the switch is closed, the galvanometer needle deflects in one direction and then returns to zero. At the instant the switch is opened, the needle deflects in the opposite direction and again returns to zero.

Figure 4. Faraday’s experiment. When the switch in the primary circuit is closed, the ammeter in the secondary circuit deflects momentarily. The emf induced in the secondary circuit is caused by the changing magnetic field through the secondary coil.


Finally, the galvanometer reads zero when there is either a steady current or no current in the primary circuit. The key to understanding what happens in this experiment is to first note that when the switch is closed, the current in the primary circuit produces a magnetic field in the region of the circuit, and it is this magnetic field that penetrates the secondary circuit. Furthermore, when the switch is closed, the magnetic field produced by the current in the primary circuit changes from zero to some value over some finite time, and this changing field induces a current in the secondary circuit.

As a result of these observations, Faraday concluded that an electric current can be induced in a circuit (the secondary circuit in our setup) by a changing magnetic field. The induced current exists for only a short time while the magnetic field through the secondary coil is changing. Once the magnetic field reaches a steady value, the current in the secondary coil disappears. In effect, the secondary circuit behaves as though a source of emf were connected to it for a short time. It is customary to say that an induced emf is produced in the secondary circuit by the changing magnetic field.
The experiments shown in Figures 3 and 4 have one thing in common: in each case, an emf is induced in the circuit when the magnetic flux through the circuit changes with time. In general,

The emf induced in a circuit is directly proportional to the time rate of change of the magnetic flux through the circuit.

This statement, known as Faraday’s law of electromagnetic induction, can be written

Faraday’s Law

$\ E = -N \frac {d\Phi_{B}}{dt}$

where $\Phi_{B} =\int \mathbf{B}\cdot d\mathbf{A}$ is the magnetic flux through the circuit in N loops. 


Lenz’s Law
The SI unit for the induced emf is the volt, V. The minus sign in the above Faraday’s law of induction is due to the fact that the induced emf will always oppose the change. It is also known as the Lenz’s law and it is stated as follows,

Lenz’s Law

The current from the induced emf will produce a magnetic field, which will always oppose the original change in the magnetic flux.


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Two Bar Magnet Magnetic Field Lines Simulation

Quiz


Q1) The force between two unlike magnetic poles

A. increases linearly with the separation distance between them
B. increases as the square of the separation distance between them
C. decreases as the separation distance between them
D. decreases as the square of the separation distance between them

Answer) D.


Q2)  An iron rod becomes magnetic when.

a. positive ions accumulate at _me end and negative ions at the other end.
b. its atoms are aligned having plus charges on one side and negative charges on the other.
c. the net spins of its electrons are in the same direction.
d. its electrons stop moving and point in the same direction.
e. none of these.

 Answer) C.

Two Bar Magnet Magnetic Field Lines Simulation(virtual experiment)


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Magnetic Fields Around Permanent Magnets
When you experimented with two magnets, you noticed that the forces between magnets, both attraction and repulsion, occur not only when the magnets touch each other, but also when they are held apart. In the same way that long-range electric and gravitational forces can be described by electric and gravitational fields, magnetic forces can be described by the existence of fields around magnets. These magnetic fields are vector quantities that exist in a region in space where a magnetic force occurs.

 Figure 1 The magnetic field of a bar magnet shows up clearly in three dimensions when the magnet is suspended in glycerol with iron filings (a). It is, however, easier to set up a magnet on a sheet of paper covered with iron filings to see the pattern in two dimensions (b).

The presence of a magnetic field around a magnet can be shown using iron filings. Each long, thin, iron filing becomes a small magnet by induction.
Just like a tiny compass needle, the iron filing rotates until it is parallel to the magnetic field. Figure 1a shows filings in a glycerol solution surrounding a bar magnet. The three-dimensional shape of the field is visible. In Figure 1b, the filings make up a two-dimensional plot of the field, which can help you visualize magnetic field lines. Filings also can show how the field can be distorted by an object. 


Magnetic field lines 
Note that magnetic field lines, like electric field lines, are imaginary. They are used to help us visualize a field, and they also provide a measure of the strength of the magnetic field. The number of magnetic field lines passing through a surface is called the magnetic flux. The flux per unit area is proportional to the strength of the magnetic field. As you can see in Figure 1, the magnetic flux is most concentrated at the poles; thus, this is where the magnetic field strength is the greatest. The direction of a magnetic field line is defined as the direction in which the north pole of a compass points when it is placed in the magnetic field. Outside the magnet, the field lines emerge from the magnet at its north pole and enter the magnet at its south pole, as illustrated in Figure 2. What happens inside the magnet? 

 Figure 2 Magnetic field lines can be visualized as closed loops leaving the north pole of a magnet and entering the south pole of the same magnet.


There are no isolated poles on which field lines can start or stop, so magnetic field lines always travel inside the magnet from the south pole to the north pole to form closed loops.

Figure 3 The magnetic field lines indicated by iron filings on paper clearly show that like poles repel (a) and unlike poles attract (b). The iron filings do not form continuous lines between like poles. Between a north and a south pole, however, the iron filings show that field lines run directly between the two magnets.

What kinds of magnetic fields are produced by pairs of bar magnets? You can visualize these fields by placing two magnets on a sheet of paper, and then sprinkling the paper with iron filings. Figure 3a shows the field lines between two like poles. In contrast, two unlike poles (north and south) placed close together produce the pattern shown in Figure 3b. The filings show that the field lines between two unlike poles run directly from one magnet to the other. 

Forces on objects in magnetic fields 
Magnetic fields exert forces on other magnets. The field produced by the north pole of one magnet pushes the north pole of a second magnet away in the direction of the field line. The force exerted by the same field on the south pole of the second magnet is attractive in a direction opposite the field lines. The second magnet attempts to line up with the field, just like a compass needle.
When a sample made of iron, cobalt, or nickel is placed in the magnetic field of a permanent magnet, the field lines become concentrated within the sample. Lines leaving the north pole of the magnet enter one end of the sample, pass through it, and leave the other end. Thus, the end of the sample closest to the magnet’s north pole becomes the sample’s south pole, and the sample is attracted to the magnet.

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A Bar Magnet Magnetic Field Lines Simulation(Virtual Experiment)

Quiz



Q1) The magnetic field lines of a bar magnet

A. begin on a north pole and end on a north pole
B. begin on a south pole and end on a south pole
C. begin on a north pole and end on a south pole
D. begin on a south pole and end on a north pole

Answer) C.



Q2)  When I cut a magnet into two pieces I get

A. An isolated north and south magnetic pole
B. Two smaller magnets
C. The two pieces are no longer magnets

 Answer) B.


Bar Magnet Magnetic Field Lines Simulation(Virtual Experiment)


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Magnetic Fields
Place a sheet of paper over a bar magnet and sprinkle iron filings on the paper. The filings will tend to trace out an orderly pattern of lines that surround the magnet. The space around a magnet, in which a magnetic force is exerted, is filled with a magnetic field. The shape of the field is revealed by magnetic field lines. Magnetic field lines spread out from one pole, curve around the magnet, and return to the other pole, as shown in Figure 1.

FIGURE 1. Iron filings trace out a pattern of magnetic field lines in the space surrounding the magnet.


Magnetic Field lines
Magnetic fields exist in three dimensions surrounding a magnet and are more intense at the poles. Magnetic fields are invisible, but can be represented in diagrams with magnetic field lines. Magnetic field lines

• point from the north pole to the south pole outside a magnet, and from the south pole to the north pole inside a magnet
• never cross one another
• are closer together where the magnetic field is stronger


Drawing Field lines
A compass can be used to map the direction of the field lines around a magnet. The compass needle will align itself along the direction of the field. Figure 2 shows the field lines around different magnets.

Figure 2 (a) A magnetic field around a bar magnet with mini compasses on the field lines, (b) A magnetic field around a horseshoe magnet with mini compasses on the field lines, (c) A magnetic field around Earth with mini compasses on the field lines


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Complex Resistor Combinations Simulation

Quiz



Q1) You have three 100 resistors available. How would you connect these three resistors to produce a 150 equivalent resistance? You must use all of the resistors.

A. Connect all three resistors in parallel.
B. Connect all three resistors in series.
C. Connect one resistor in series with two resistors in parallel.
D. Connect two resistors in series with the third resistor in parallel to the first two.

Answer) C.



Q2)  A circuit is constructed as follows: four resistors in parallel connected in series with three resistors in parallel connected in series with two resistors in parallel. All of the resistors have the same value. How does the equivalent resistance of this circuit compare to the resistance of a single resistor?

A. The equivalent resistance is less than a single resistor.
B. The equivalent resistance is the same as a single resistor.
C. The equivalent resistance is greater than a single resistor.
D. The equivalent resistance cannot be determined without the resistance value of a single resistor.

 Answer) C.


Resistors in parallel total resistance Simulation


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RESISTORS COMBINED BOTH IN PARALLEL AND IN SERIES
Series and parallel circuits are not often encountered independent of one another. Most circuits today employ both series and parallel wiring to utilize the advantages of each type.
A common example of a complex circuit is the electrical wiring typical in a home. In a home, a fuse or circuit breaker is connected in series to numerous outlets, which are wired to one another in parallel. An example of a typical household circuit is shown in Figure 1.

Figure 1. (a) When all of these devices are plugged into the same household circuit, (b) the result is a parallel combination of resistors in series with a circuit breaker.

As a result of the outlets being wired in parallel, all the appliances operate independently; if one is switched off, any others remain on.Wiring the outlets in parallel ensures that an identical potential difference exists across any appliance.
This way, appliance manufacturers can produce appliances that all use the same standard potential difference.
To prevent excessive current, a fuse or circuit breaker must be placed in series with all of the outlets. Fuses and circuit breakers open the circuit when the current becomes too high. A fuse is a small metallic strip that melts if the current exceeds a certain value. After a fuse has melted, it must be replaced. A circuit breaker, a more modern device, triggers a switch when current reaches a certain value. The switch must be reset, rather than replaced, after the circuit overload has been removed. Both fuses and circuit breakers must be in series with the entire load to prevent excessive current from reaching any appliance. In fact, if all the devices in Figure 1 were used at once, the circuit would be overloaded. The circuit breaker would interrupt the current.

Fuses and circuit breakers are carefully selected to meet the demands of a circuit. If the circuit is to carry currents as large as 30 A, an appropriate fuse or circuit breaker must be used. Because the fuse or circuit breaker is placed in series with the rest of the circuit, the current in the fuse or circuit breaker is the same as the total current in the circuit. To find this current, one must determine the equivalent resistance.
When determining the equivalent resistance for a complex circuit, you must simplify the circuit into groups of series and parallel resistors and then find the equivalent resistance for each group by using the rules for finding the equivalent resistance of series and parallel resistors.

Work backward to find the current in and potential difference across a part of a circuit Now that the equivalent resistance for a complex circuit has been determined, you can work backward to find the current in and potential difference across any resistor in that circuit. In the household example, substitute potential difference and equivalent resistance in $\Delta V = IR$ to find the total current in the circuit. Because the fuse or circuit breaker is in series with the load, the current in it is equal to the total current. Once this total current is determined, $\Delta V = IR$ can again be used to find the potential difference across the fuse or circuit breaker.

There is no single formula for finding the current in and potential difference across a resistor buried inside a complex circuit. Instead, $\Delta V = IR$ and the rules reviewed in Table must be applied to smaller pieces of the circuit until the desired values are found.


Resistors in parallel equivalent resistance Simulation

Quiz



Q1) A current of 5 ampere flows into a circuit where two identical resistors are attached in parallel. What current would flow through one of the resistors?

A. Both resistors carry 5 A
B. each resistor will carry 2.5 A
C. It is impossible to tell unless you know the size of the resistance of each resistor
D. The resistor would carry less than 5 A but you cant tell how much energy was lost before

Answer) B.


Q2)  Four resistors having equal values are wired as a parallel circuit. How does the equivalent resistance of the circuit compare to the resistance of a single resistor?

A. The equivalent resistance is greater than the resistance of any single resistor.
B. The equivalent resistance is the same as the resistance of any single resistor.
C. The equivalent resistance is one-fourth the resistance value of a single resistor.
D. The equivalent resistance is one-half the resistance value of a single resistor.

 Answer) C.

Resistors in parallel equivalent resistance Simulation


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RESISTORS IN PARALLEL
As discussed above, when a single bulb in a series light set burns out, the entire string of lights goes dark because the circuit is no longer closed.What would happen if there were alternative pathways for the movement of charge, as shown in Figure 1.
A wiring arrangement that provides alternative pathways for the movement of a charge is a parallel arrangement. The bulbs of the decorative light set shown in the schematic diagram in Figure 1 are arranged in parallel with each other.

Figure 1. These decorative lights are wired in parallel. Notice that in a parallel arrangement there is more than one path for current.


Resistors in parallel have the same potential differences across them 
To explore the consequences of arranging resistors in parallel, consider the two bulbs connected to a battery in Figure 13(a). In this arrangement, the left side of each bulb is connected to the positive terminal of the battery, and the right side of each bulb is connected to the negative terminal. Because the sides of each bulb are connected to common points, the potential difference across each bulb is the same. If the common points are the battery’s
terminals, as they are in the figure, the potential difference across each resistor is also equal to the terminal voltage of the battery. The current in each bulb, however, is not always the same.

Figure 2. (a) This simple parallel circuit with two bulbs connected to a battery can be represented by (b) the schematic diagram shown on the right.

The sum of currents in parallel resistors equals the total current 
In Figure 2, when a certain amount of charge leaves the positive terminal and reaches the branch on the left side of the circuit, some of the charge moves through the top bulb and some moves through the bottom bulb. If one of the bulbs has less resistance, more charge moves through that bulb because the bulb offers less opposition to the flow of charges.
Because charge is conserved, the sum of the currents in each bulb equals the current $\ I$ delivered by the battery. This is true for all resistors in parallel.

$\ I =  I_{1} + I_{2} + I_{3}.....$

The parallel circuit shown in Figure 2 can be simplified to an equivalent resistance with a method similar to the one used for series circuits. To do this, first show the relationship among the currents.

$\ I =  I_{1} + I_{2}$

Then substitute the equivalents for current according to $\Delta V = IR$.

$\frac{\Delta V}{R_{eq}}=\frac{\Delta V_{1}}{R_{1}} + \frac{\Delta V_{2}}{R_{2}}$

Because the potential difference across each bulb in a parallel arrangement equals the terminal voltage ($\Delta V$ = $\Delta V_{1}$ = $\Delta V_{2}$), you can divide each side of the equation by $\Delta V$ to get the following equation.


$\frac{1}{R_{eq}}=\frac{1}{R_{1}} + \frac{1}{R_{2}}$

An extension of this analysis shows that the equivalent resistance of two or more resistors connected in parallel can be calculated using the following equation.


RESISTORS IN PARALLEL
$\frac{1}{R_{eq}}=\frac{1}{R_{1}} + \frac{1}{R_{2}}$.....
The equivalent resistance of resistors in parallel can be calculated using a reciprocal relationship.

Notice that this equation does not give the value of the equivalent resistance directly. You must take the reciprocal of your answer to obtain the value of the equivalent resistance.
Because of the reciprocal relationship, the equivalent resistance for a parallel arrangement of resistors must always be less than the smallest resistance in the group of resistors.

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Resistors in series total resistance Simulation

Quiz



Q1) When three resistors are combined in series the total resistance of the combination is

A. greater than any of the individual resistance values.
B. less than any of the individual resistance values.
C. the average of the individual resistance values.

Answer) A.



Q2)  Several resistors are wired in series. What is true about this circuit?

A. The sum of the currents through each of the resistors is equal to the total circuit current.
B. The total circuit current is the same as the current through any one of the resistors.
C. The voltage across any resistor is the same as the voltage of the power supply.
D. The current in any single resistor is determined by its resistance and the voltage of the power supply.


 Answer) B.


Resistors in series total resistance Simulation


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RESISTORS IN SERIES 
In a circuit that consists of a single bulb and a battery, the potential difference across the bulb equals the terminal voltage. The total current in the circuit can be found using the equation $\Delta V = IR$. 

What happens when a second bulb is added to such a circuit, as shown in Figure 1. When moving through this circuit, charges that pass through one bulb must also move through the second bulb. Because all charges in the circuit must follow the same conducting path, these bulbs are said to be connected in series.


Resistors in series carry the same current 
Light-bulb filaments are resistors; thus, Figure 1(b) represents the two bulbs in Figure 1(a) as resistors. Because charge is conserved, charges cannot build up or disappear at a point. For this reason, the amount of charge that enters one bulb in a given time interval equals the amount of charge that exits that bulb in the same amount of time. Because there is only one path for a charge to follow, the amount of charge entering and exiting the first bulb must equal the amount of charge that enters and exits the second bulb in the same time interval. 

Because the current is the amount of charge moving past a point per unit of time, the current in the first bulb must equal the current in the second bulb. This is true for any number of resistors arranged in series. When many resistors are connected in series, the current in each resistor is the same. 

The total current in a series circuit depends on how many resistors are present and on how much resistance each offers. Thus, to find the total current, first use the individual resistance values to find the total resistance of the circuit, called the equivalent resistance. Then the equivalent resistance can be used to find the current.

Figure 1. These two light bulbs are connected in series. Because light-bulb filaments are resistors, (a) the two bulbs in this series circuit can be represented by (b) two resistors in the schematic diagram shown on the right.


The equivalent resistance in a series circuit is the sum of the circuit’s resistances
As described in Section 1, the potential difference across the battery, ΔV, must equal the potential difference across the load, $\Delta V_{1}$ + $\Delta V_{2}$ , where $\Delta V_{1}$ is the potential difference across $\ R_{1}$ and $\Delta V_{2}$ is the potential difference across $\ R_{2}$.

 $\Delta V = \Delta V_{1} +  \Delta V_{2}$

According to $\ V = IR$, the potential difference across each resistor is equal to the current in that resistor multiplied by the resistance.

 $\Delta V = I_{1} R_{1} + I_{2} R_{2}$

Because the resistors are in series, the current in each is the same. For this reason, $\ I_{1}$and $\ I_{2}$ can be replaced with a single variable for the current, $\ I$.

 $\Delta V = I ( R_{1} + R_{2})$


Figure 2. (a) The two resistors in the actual circuit have the same effect on the current in the circuit as (b) the equivalent resistor.

Finding a value for the equivalent resistance of the circuit is now possible.
If you imagine the equivalent resistance replacing the original two resistors, as shown in Figure 2, you can treat the circuit as if it contains only one resistor and use $\Delta V = IR$ to relate the total potential difference, current, and equivalent resistance.

 $\Delta V = I(R_{eq})$

Now set the last two equations for V equal to each other, and divide by the current.

 $\Delta V = I(R_{eq}) = I(R_{1} + R_{2})$
$\ R_{eq} = R_{1} + R_{2}$

Thus, the equivalent resistance of the series combination is the sum of the individual resistances. An extension of this analysis shows that the equivalent resistance of two or more resistors connected in series can be calculated using the following equation.


RESISTORS IN SERIES
$\ R_{eq} = R_{1} + R_{2} + R_{3} . . .$
Equivalent resistance equals the total of individual resistances in series.


Because $\ R_{eq} $ represents the sum of the individual resistances that have been connected in series, the equivalent resistance of a series combination of resistors is always greater than any individual resistance.
To find the total current in a series circuit, first simplify the circuit to a single equivalent resistance using the boxed equation above; then use $\Delta V = IR$ to calculate the current.

$\ I=\frac{\Delta V}{R_{eq}}$

Because the current in each bulb is equal to the total current, you can also use  $\Delta V = IR$  to calculate the potential difference across each resistor.

 $\Delta V_{1} = IR_{1}$ and $\Delta V_{2} = IR_{2}$ 

The method described above can be used to find the potential difference across resistors in a series circuit containing any number of resistors.


Series circuits require all elements to conduct
What happens to a series circuit when a single bulb burns out? Consider what a circuit diagram for a string of lights with one broken filament would look like. As the schematic diagram in Figure 3 shows, the broken filament means that there is a gap in the conducting pathway used to make up the circuit. Because the circuit is no longer closed, there is no current in it and all of the bulbs go dark.

Figure 3. A burned-out filament in a bulb has the same effect as an open switch. Because this series circuit is no longer complete, there is no current in the circuit.


Why, then, would anyone arrange resistors in series? Resistors can be placed in series with a device in order to regulate the current in that device. In the case of decorative lights, adding an additional bulb will decrease the current in each bulb. Thus, the filament of each bulb need not withstand such a high current.
Another advantage to placing resistors in series is that several lesser resistances can be used to add up to a single greater resistance that is unavailable. Finally, in some cases, it is important to have a circuit that will have no current if any one of its component parts fails. This technique is used in a variety of contexts, including some burglar alarm systems.

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Electrostatic Induction Simulation of Electroscope

Quiz



Q1) When you brush your hair and scrape electrons from your hair, the charge of your hair is

A. positive.
B. negative.
C. both A and B
D. neither A nor B

Answer) A.
And if electrons were scraped off the brush onto your hair, your hair would have a negative charge.


Q2) How does charging an object by contact differ from charging an object by induction?

 A. An object charged by contact gains the same type of charge while an object charged by induction gains the opposite type of charge.
 B. An object charged by contact is touched while an object charged by induction is not.
 C. An object charged by contact must be a conductor while an object charged by induction is an insulator.

Answer) A.


Electrostatic Induction Simulation of Electroscope


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Electrically Charged Objects 
Matter is made of atoms, and atoms are made of electrons and protons (and neutrons as well). An object that has equal numbers of electrons and protons has no net electric charge. But if there is an imbalance in the numbers, the object is then electrically charged. An imbalance comes about by adding or removing electrons.
Although the innermost electrons in an atom are bound very tightly to the oppositely charged atomic nucleus, the outermost electrons of many atoms are bound very loosely and can be easily dislodged. 

FIGURE 01. When electrons are transferred from the fur to the rod, the rod becomes negatively charged.

How much energy is required to tear an electron away from an atom varies for different substances. The electrons are held more firmly in rubber than in fur, for example. Hence, when a rubber rod is rubbed by a piece of fur, as illustrated in Figure 01, electrons transfer from the fur to the rubber rod. The rubber then has an excess of electrons and is negatively charged. The fur, in turn, has a deficiency of electrons and is positively charged. If you rub a glass or plastic rod with silk, you’ll find that the rod becomes positively charged. The silk has a greater affinity for electrons than the glass or plastic rod. Electrons are rubbed off the rod and onto the silk.
An object that has unequal numbers of electrons and protons is electrically charged. If it has more electrons than protons, the object is negatively charged. If it has fewer electrons than protons, it is positively charged.


The TriboElectric Series
If we did a study of many materials and put them in order from those with the least desire for  electrons to those with a very strong desire for electrons we would have created a Triboelectric series. If two items from the list are rubbed together, then the item that is higher on the list will end up more positive and the lower one will end up more negatively charged.  For example, if leather were rubbed with wool, the leather becomes positive and the wool negative.  Yet if rubber is rubbed with wool, the rubber becomes negative and the wool positive.
It is important to note that this series is true only if the samples are clean and dry.  The presence of moisture, dirt, or oils may cause some of the items to interact differently.

Table 01. TRIBOELECTRIC SERIES
Positive(+)
Air
Human Hands
Asbestos
Rabbit's Fur
Glass
Human Hair
Mica
Nylon
Wool
Lead
Cat's Fur
Silk
Aluminum
Paper
Cotton
Steel
Wood
Lucite
Sealing wax
Amber
Polystyrene
Polyethylene
Rubber balloon
Sulphur
Hard rubber
Nickel, Copper
Brass, Silver
Gold, Platinum
Sulfur
Acetate, Rayon
Polyester
Celluloid
Polyurethane
Polyethylene
Polypropylene
Vinyl
Silicon
Teflon
Saran Wrap
 Negative(-)

Forces on Charged Bodies
The forces that you observed on tape strips also can be demonstrated by suspending a negatively charged, hard rubber rod so that it turns easily, as shown in Figure 02. If you bring another negatively charged rod near the suspended rod, the suspended rod will turn away. The negative charges on the rods repel each other. It is not necessary for the rods to make contact.

Figure 02. A charged rod, when brought close to another charged and suspended rod, will attract or repel the suspended rod.

The force, called the electric force, acts at a distance. If a positively charged glass rod is suspended and a similarly charged glass rod is brought close, the two positively charged rods also will repel each other. If a negatively charged rod is brought near a positively charged rod, however, the two will attract each other, and the suspended rod will turn toward the oppositely charged rod. The results of your tape experiments and these actions of charged rods can be summarized in the following way:

• There are two kinds of electric charges: positive and negative.
• Charges exert forces on other charges at a distance.
• The force is stronger when the charges are closer together.
• Like charges repel; opposite charges attract.

Neither a strip of tape nor a large rod that is hanging in open air is a very sensitive or convenient way of determining charge. Instead, a device called an electroscope is used. An electroscope consists of a metal knob connected by a metal stem to two thin, lightweight pieces of metal foil, called leaves. Figure 03 shows a neutral electroscope. Note that the leaves hang loosely and are enclosed to eliminate stray air currents.

Figure 03. An electroscope is a device used for detecting charges. In a neutral electroscope, the leaves hang loosely, almost touching one another.

Charging by conduction 
When a negatively charged rod is touched to the knob of an electroscope, electrons are added to the knob. These charges spread over all the metal surfaces. As shown in Figure 04a, the two leaves are charged negatively and repel each other; therefore, they spread apart.
The electroscope has been given a net charge. Charging a neutral body by touching it with a charged body is called charging by conduction. The leaves also will spread apart if the electroscope is charged positively. How, then, can you find out whether the electroscope is charged positively or negatively? The type of charge can be determined by observing the leaves when a rod of known charge is brought close to the knob. The leaves will spread farther apart if the rod and the electroscope have the same charge, as shown in Figure 04b. The leaves will fall slightly if the electroscope’s charge is opposite that of the rod, as in Figure 04c.


Figure 04. A negatively charged electroscope will have its leaves spread apart (a). A negatively charged rod pushes electrons down to the leaves, causing them to spread farther apart (b). A positively charged rod attracts some of the electrons, causing the leaves to spread apart less (c).


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