|
| A positive charge means that the object has lost electrons and is no longer electrically neutral. Each electron lost gives the particle a charge of +1.6 x 10-19 coulombs. Positive, or vitreous, charges are classically created by rubbing a glass rod with silk. The rod becomes positive (loses electrons); the silk become negative (gains electrons). Since electric charge is conserved, the system (glass rod and silk) maintains a net charge of 0. A negative charge means that the object has gained electrons. Each electron gained gives the particle an additional charge of -1.6 x 10-19 coulombs. Negative, or resinous, charges are classically created by rubbing a rubber rod with fur. The rod becomes negatively charged; the fur positively charged. By definition, negatively charged objects have more mass than an identical neutral object since each extra electron has a mass of 9.11 x 10-31 kg. Three modes of electrifying an object: friction, conduction and induction. Electrification by friction occurs when two surfaces are rubbed together. Examples of this were discussed above when a positive charge was created by rubbing glass with silk and a negative charge was created by rubbing rubber with fur. The following list details a larger portion of the triboelectric sequence. When any two substances shown in this list are rubbed together, the top one will become positively charged while the lower one will become negatively charged. The further apart the two substances are in the list, the greater the electrification. | +
- | Asbestos
Fur (rabbit)
Glass
Mica
Wool
Quartz
Fur (cat)
Lead
Silk
Human skin, Aluminum
Cotton
Wood
Amber
Copper, Brass
Rubber
Sulfur
Celluloid
India rubber |
Charging by conduction means that the charging rod actually touches the electroscope?s knob. Since there is contact, electrons from the knob would flow onto a positive rod or off of a negative rod. Charging by conduction leaves the electroscope with a residual charge IDENTICAL to that of the charging rod.
Charging by induction means that the charging rod is brought close to the electroscope?s knob but NEVER touches it. If the electroscope is not grounded, it will remain neutral but be temporarily polarized while the charging rod is in the immediate vicinity. That is, a positive rod will induce the electrons in the scope to migrate to the knob. This redistribution of charge will result in the leaves of the scope being positively charged If the electroscope is grounded during induction, electrons will flow from the knob to the ground if the charging rod is NEGATIVE and electrons will flow onto the knob if the charging rod is POSITIVE. The net effect once the grounding wire is removed is that the electroscope will be left with a residual charge that is OPPOSITE to that of the charging rod. Suppose the positive rod is brought near to an insulator, for example, a piece of paper or a section of a wall. Since electrons are not free to move within an insulator, another process takes place which still results in the paper or wall becoming polarized. The particles in the insulator realign themselves - presenting an oppositely charged layer towards the charged rod. This process is illustrated below.  | positively charged rod | | top surface "-" | | polarized molecules
within the insulator | | bottom surface "+" |
Coulombs Law
Coulomb's Law: calculating electrostatic forces between point charges
Coulomb?s Law of Electrostatics states that
 where:
F is the force measured in Newtons
k is Coulomb's constant which equals 9 x 109 Nm2/C2
Q is the magnitude of each charge measured in coulombs
r is the distance between the centers of the two charges
This formula may only be used for point charges. That is, for isolated points of electric charge. It is an example of an inverse square law: if you double the distance between two charges, the force between them is reduced to 1/4th its original size. - If F is negative, that means that the charges carry opposite "signs" -- that is, one is positive and the other is negative. The negative answer means that the point charges are attracting each other -- it does NOT mean that F is acting in a negative direction.
- If F is positive, that means that the charges carry the same "sign" -- that is, either both are positive or both are negative. A positive answer means that the point charges are repelling each other -- it does NOT mean that F is acting in a positive direction.
| Refer to the following information for the next four questions. Two identical conducting spheres are initially charged and separated at a distance of 1 meter, as shown below. | |
Summary: Take a moment to compare the properties of gravitational forces with those of electrostatic forces.
| Gravitational Forces | Electrostatic Forces | - G = 6.67 x 10-11 Nm2/kg2 is VERY small
- gravity is a weak force
- inverse square force
- attractive only
| - k = 9 x 109 Nm2/C2 is VERY large
- electrostatic forces are strong
- inverse square force
- attractive and repulsive
| | | |
Electric Field
| An electric field is the region surrounding a charged particle, Q, where another charged particle with experience either a force of attraction or repulsion. For point charges, the electric field lines are radial, getting ever farther apart as you get farther from the point charge itself. These fields are NOT uniform, but are examples of inverse square fields E = kQ/r2. Dimensional analysis reveals that the units on E, the electric field strength, should be E = kQ /r2
(Nm2/C2)(C)(1/m2)
N/C Intuitively, the electric field strength measures the amount of force, in newtons, experienced by a coulomb of charge when it is placed at a particular position within an electric field. Electric field lines point in the direction in which a positive test charge would respond to the electrostatic force; that is, away from positive charges and towards negative charges. In the following diagram, Q is positive, since the field lines are pointing away from Q. If Q had been negative, then the field lines would have pointed towards Q. Note that field lines are NEVER allowed to cross each other. Another property of field lines is that they terminate on the surface of a charge - they do not penetrate into the charge. Consequently, there is no electric field within a charged conductor under electrostatic conditions. This fact is illustrated in the diagram of E vs r where it shows that the magnitude of the electric field equals 0 between 0 and r. The size of two charges can be compared by noting the relative number of field lines surrounding each one. If a second charge with only 8 fields lines was compared to the diagram provided above, then it would indicate that the second charge was only ½ as large, since 8 is half of 16. Fields between oppositely charged particles are attractive and are elliptical in shape; while fields between similarly charged particles are repulsive and hyperbolic in shape.  |  | | oppositely charged particles
left is positive; right is negative | similarly charged particles
both are positive | A convenient way to remember the properties of an electric field are to use analogies to gravitational fields. A gravitational field is the region surrounding a massive object in which another object with mass will experience a force of gravitational attraction. One important distinction between electrical fields and gravitational fields is that electrical fields can be both attractive and repulsive; whereas gravitational fields are only attractive. Subsequently, gravitational fields cannot be shielded. | Gravitational Forces | Electrostatic Forces | - G = 6.67 x 10-11 Nm2/kg2 is VERY small
- gravity is a weak force
- inverse square force
- attractive only
| - k = 9 x 109 Nm2/C2 is VERY large
- electrostatic forces are strong
- inverse square force
- attractive and repulsive
| | Gravitational Field | Electrostatic Field (+ charge) | - gravitational field strength, g
- g's vector nature points towards the center of the planet
- each surface represents a unique value for g
- g is measured in N/kg (or m/sec2)
| - electric field strength, E
- E's vector nature points away from a positive charge or towards a negative charge
- each surface represents a unique value for E
- E is measured in N/C
| In the chart above, you can see that both fields are inverse square relationships. That the electric field strength, E, has the same configuration as the gravitational field strength, g. That in each case, the force experienced by a second object equals the product of either that object's mass times the gravitational field strength or that object's charge times the electrical field strength. The direction of the gravitational field is defined as the direction a second object with mass would be attracted; whereas the direction of an electrical field is defined as the direction a positive test charge would respond. | Electric Potential | When charged particles are moved from one position in an electric field to another position, a new unit of measurement is needed. A volt represents the amount of work per unit charge required to move a charge between two positions in an electric field. If it takes 1 joule of work to move 1 coulomb of charge between two positions in an electric field, then those positions have a potential difference of 1 volt. Voltage is a scalar property of an electric field, it has no direction, only magnitude. In general, 1 volt = 1 joule / 1 coulomb
Rearranging these units (1 joule = 1 coulomb x 1 volt) shows us that the amount of work done on a charge by an external agent as it is moved around an electric field is expressed as
Wexternal = qV For a point charge the absolute potential of any position in its electric field can be calculated using the equation Vabs = kQ/r When the charge creating the field is positive, the voltage is positive; when the central charge is negative, the voltage is negative. As r grows larger and larger, that is, as r approaches infinity, the absolute potential is defined to be zero. You can almost think of the "voltage" as being an indicator of the "elevation of the terrain" surrounding a point charge. The steeper the terrain, the faster the voltage changes from one location to another. The work done by an external agent can be envisioned as "pushing or pulling" a second charge up or down these changes in elevation. | | voltage "profile" - charge | | voltage "profile" + charge | | | Refer to the following information for the next three questions. The central charge, Q, has a charge of 10 µC. | Surfaces which connect points that are at the same absolute potential, or voltage, are called equipotential surfaces. In the diagram of the point charge shown in the previous example, two equipotential surfaces were labeled, A and B. Notice that equipotential surfaces meet field lines at right angles. The closer together two equipotential surfaces are to each other, the more rapid the change in voltage. This indicates a stronger electric field which is shown in the second diagram below by the fact that the field lines are grouped closer together on the left side than on the right. Note that the electric field strength, E, can be measured in either the units V/m, or equivalently, in the unit N/C. N/C = V / d
= (J/C) / m
= [(Nm)/C] / m = N/C The following two graphs compare the voltage around a positively charged conducting sphere and the electric field for a positively charged conducting sphere. Note that the electric field strength (E ? 1/r2) drops off more rapidly than does the voltage (V ? 1/r). Also notice that within a conducting sphere, the voltage remains constant in contrast to the fact that no electric field exists. | | | | For a conducting sphere,
V = kQ/r | For a conducting sphere,
E = kQ/r2 | Remember that the electric field strength, E, is a vector quantity. You are required to state both its magnitude and its direction to completely describe it at any given location. If you are ever asked to calculate the net electric field in 2-dimensions, you should first take the x- and y-components of each field, add the components to determine the net Ex and net Ey, and then calculate the resultant field and its direction. Voltage, on the other hand, is a scalar quantity and can be added directly without considering components or directions. Let's work through the next examples to show you the difference in these two field properties. In each set of diagrams, compare the charge configuration diagram and voltage diagram to determine the requested information for each midpoint. The charges are assumed to be a distance "r" apart. | | Charge Configuration | Voltage Diagram |  |  | | Charge Configuration | Voltage Diagram |  |  | | Charge Configuration | Voltage Diagram |  |  | | Charge Configuration | Voltage Diagram |  |  | Suppose two conductive spheres are both charged to +6 µC. The smaller sphere has a radius of 0.5 cm while the larger sphere has a radius of 1.5 cm. . The spheres are now connected with a conducting wire. Conductor | What is a conductor? Conductors are materials, for example, metals, through which charged particles move readily. What is meant by the term "under electrostatic conditions?" - Under electrostatic conditions, field lines must terminate or begin on the surface of a conductor - that is, there is no electric field within a conductor. If any field line penetrated into the conductor, then electrons would respond to its presence and be accelerated within the conductor. If that happened, the conductor would no longer be under electrostatic conditions.
- Under electrostatic conditions, field lines must meet the surface of a conductor at right angles. If any field line did NOT come it at a 90º angle, then a component of the field line would be parallel to the conductor?s surface and electrons would respond to its presence and be accelerated within the conductor. If that happened, the conductor would no longer be under electrostatic conditions.
When a conductor is under electrostatic conditions, all charges (electrons) must be at rest. Don't forget that one coulomb of charge represents 6.25 x 1018 electrons. - Under electrostatic conditions, the entire conductor must be at the same potential, or voltage. If not, then charges would flow from points of high potential to points of lower potential. If that happened, the conductor would no longer be under electrostatic conditions.
Faraday's Ice-Pail Experiment  - Faraday started with a neutral metal ice pail (metal bucket) and an uncharged electroscope.
- He then suspended a positively charged metal ball into the ice pail, being careful to not touch the sides of the pail. The leaves of the electroscope diverged. Moreover, their degree of divergence was independent of the metal ball's exact location. Only when the metal ball was completely withdrawn did the leaves collapse back to their original position.
- Faraday noticed that if the metal ball was allowed to contact the inside surface of the ice pail, the leaves of the electroscope remained diverged.
- Afterwards, when he completely removed the ball from the inside of the ice pail, the leaves remained diverged. However, the metal ball was no longer charged. Since the leaves of the electroscope that was attached to the OUTSIDE of the pail did not move when the ball touched the inside of the pail, he concluded that the inner surface had just enough charge to neutralize the ball.
Conclusions: Faraday's Ice Pail Experiment
pg 330-331, Principles of Physics, Frederick Beuche, McGraw-Hill Book Company, New York, New York. 1988. - A charged metal object suspended inside a neutral metal container INDUCES an equal but opposite charge on the inside of the container.
- When the charged metal object is touched to the inside of the of the container, the induced charge exactly neutralizes the excess charge on the object.
- When a charged object is placed within a metal container, an equal charge of the same sign is FORCED to the outer surface of the container.
- All of the charge on any metal object resides on its outer surface if a conducting path is provided so that the charge can move there. Remember that charges will flow between two positions as long as there is a potential difference between those positions. When the voltage has been equalized, all charges will cease to flow.
Faraday Cage An important consequence of this experiment is that electric fields can be shielded - that is, the outside of a conductor acts as a FARADAY CAGE. A closed metal surface, no matter what it's shape, will block out any external electric field lines. And, as long as there are no electric charges residing inside the metal cavity, the electric field in the interior will be ZERO everywhere. This is why you are safe inside your car or on an airplane during a lightning storm. Electrical shielding is easily accomplished by surrounding the surface that you wish to shield with a conducting surface. The free charges on the conducting surface will arrange themselves in such a way as to insure that the electric field within the conductor equals zero. This is the reason why electrical components come in metal boxes, to shield them from outside electrical activity. This principle of electrical shielding is an important distinction between electric fields and gravitational fields. Electric fields can be shielded since there are two (2) types of electric charges. However, gravitational fields CANNOT be shielded - the effects of the gravitational attraction between two objects can be felt through any and all intervening matter. | | | | Conducting Shells Consider the charged conducting sphere shown above. Since there are eight field lines illustrated, let's assume that its charge is +8 µC. When viewed from infinity, this charged sphere would look like a point charge with an electric field that agrees with the graph for E vs r shown above. However, consider that this charged sphere could instead be constructed of a NEUTRAL thin conducting shell with a hidden positive point charge located at its center, as shown below.  According to the results of Faraday's Ice Pail Experiment, the positive point charge INDUCES an equal but opposite charge on the inside of the shell AND an equal but similar charge on the outside of the shell. Note that the shell remains neutral - eight field lines terminate on the surface of its inner shell and eight field lines originate on the surface of its outer shell. Remember that there would be NO field lines between the "inner and outer" surfaces of the conducting shell. The only field lines would be between the inside point charge and the shell's inner surface and outside of the shell's outer surface. All field lines should be symmetric and meet any equipotential surfaces at right angles. When viewed from infinitely far away, this configuration would look exactly like the original 8 µC charged sphere!
Given below is a diagram of the electric fields for this conducting shell. For the remainder of this lesson we will work some examples using conducting shells. In each case, the conducting shell is aqua in color and the point charge placed in its center is yellow in color.
| Refer to the following information for the next three questions. Refer to the following information for the next three questions. Refer to the following information for the next three questions. Refer to the following information for the next three questions. | | When two parallel plates are connected across a battery, the plates will become charged and an electric field will be established between them. Remember that the direction of an electric field is defined as the direction that a positive test charge would move. So in this case, the electric field would point from the positive plate to the negative plate. Since the field lines are parallel to each other, this type of electric field is uniform and is calculated with the equation E = V/d. Note that the electric field strength, E, can be measured in either the units V/m, or equivalently, N/C. [E] = V/d
(J/C)/m
(Nm)/C/ m N/C Since the field lines are parallel and the electric field is uniform between two parallel plates, a test charge would experience the same force of attraction or repulsion no matter where it is located. That force is calculated with the equation F = qE. To review more about electric fields between parallel plates, go back and review this resource lesson. Capacitance When two plates are charged and used in an electric circuit, that device is called a capacitor. It's role in the circuit is to store energy. Capacitors are rated in terms of their capacitance which is measured in farads (F). One farad equals the ratio of one coulomb per volt. [F] = C/V A parallel plate capacitor's effective capacitance is defined in terms of its geometry. C = ?oA/d where ?o, the permittivity of free space, is a constant equal to 8.85 x 10-12 F/m,
A is the cross sectional area of ONE plate, and
d is the distance between the plates. Essentially, capacitance measures the relative amount of charge that can be stored on a pair of parallel plate for a given amount of voltage. If the capacitance increases, then more charge can be stored when the same potential is applied. The equation for the line becomes Q = CV and the equation for the area under the curve becomes E = ½QV = ½CV2. The plates can then be discharged later through an external circuit. They are used when the circuit requires a big burst of energy; for example: to "jump start" electric motors, TV's or operate flash attachments on a camera. | Refer to the following information for the next three questions. A parallel-plate capacitor is connected across a 9-volt battery. Each plate of the capacitor has a cross-sectional area of 0.0016 m2 and the plates are separated by 5 µm of air. | | Combinations of Capacitors When more than one capacitors are used in a circuit, the above formula is restated as Qtotal = Ctotal x Vtotal If the capacitors are arranged in series (one after another along a single path), then Qseries = Q1 = Q2 = Q3
Cseries = (1/C1 + 1/C2 + 1/C3)-1
Vseries = V1 + V2 + V3 If the capacitors arranged in parallel (strung along multiple paths that cross the same section), then Qparallel = Q1 + Q2 + Q3
Cparallel = C1 + C2 + C3
Vparallel = V1 = V2 = V3 Springs and Capacitors Let's take a moment and note a similarity between springs and capacitors.
For a simple spring, Fdistorting = ks and the energy stored is PEe = ½ks2. When springs are combined in series, the spring constant for the system becomes kseries = (1/k1 + 1/k2 + 1/k3)-1. When springs are combined in parallel, the spring constant for the system becomes kparallel = k1 + k2 + k3 These rules exactly model those of capacitors. The similarities make sense since both springs and capacitors are energy-storage devices: springs store mechanical energy; capacitors store electrical energy. Resistors and Capacitors Note that there are both similarities and differences between the rules for capacitors and resistors. When resistors are wired in series, Iseries = I1 = I2 = I3 Rseries = R1 + R2 + R3 Vseries = V1 + V2 + V3
Notice that these "circuit properties" agree with those of capacitors:
- the current (C/sec) through devices wired in series is the same, charges will "flow" between capacitors until they equalize;
- voltages changes across devices wired in series are additive, whether they are resistors or capacitors.
Similarly, for resistors wired in parallel, Iparallel = I1 + I2 + I3 Rparallel = (1/R1 + 1/R2 + 1/R3)-1 Vparallel = V1 = V2 = V3 Notice once again the agreement of these "circuit properties" with those of capacitors:
- when currents (C/sec) divide in parallel, the charges on the capacitor plates would consequently need to add together;
- the voltage across parallel devices is the same, whether they are resistors or capacitors.
However, the rules for resistance and capacitance are "reversed" since resistors are devices that dissipate energy while capacitors are devices that store energy. Practice Examples | Refer to the following information for the next six questions. Each of the following capacitors has a rated capacitance of 10 µF. Refer to the following information for the next five questions. Each of the following capacitors has a rated capacitance of 10 µF. | | |
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