Chapter 16 Physics Review
If one object or region of space acquires a positive charge, what occurs?
An equal amount of negative charge will be found in neighboring areas or objects. There is no violation or exception found.
Conductors vs Nonconductors or Insulators
Materials that are able to transfer electricity and transfer electrons between materials like metals are known as conductors. Whereas wood and rubber are nonconductors or insulators as they are not really able to conduct or transfer much if any charge between materials.
Unit for Coulombs Law and Constant
The unit for Charge is Coulomb's (C), the Unit of Force in Coulombs law is N, unit for r is meters. k constant = 8.988 * 10^9 N*m^2/C^2 or if two sig figs are used: 9 * 10^9 1 C is the amount of charge which, if placed on each of tw o point objects that are 1.0 m apart, will result in each object exerting a force of 9 * 10^9 N on each other: (9*10^9 Nm^2/C^2)(1 C) (1 C) / (1 m)^2 = 9 *10^9 N
The example associated with electric charges
There are two types of electric charge. This can be demonstrated through the following experiment. A plastic ruler suspended by a thread is vigorously rubbed with a cloth to charge it. When a second plastic ruler, which has been charged in the same way is brought close to the first, its found taht one ruler repels the other. Similarly, if a rubbed glass rod is brought close to a second charged glass rod, again a repulsive force is seen. However, if the charged glass rod is brought close to the charged plastic ruler, its found that they attract each other. The charge on the glass must therefore be different from that on the plastic. Therefore this leads us to determining that there are two types of electric charge. Each type of charge repels the same type but attracts the opposite type. That is: unlike charges attract; like charges repel.
Problem Solving in Electrostatics
1. Draw a Diagram, show all charges, with signs, and electric fields and forces with directions. 2. Calculate forces using Coulombs Law, or the magnitude of the electric field each charge produces at a given point. 3. Add forces or fields vectorially to get result.
Summary of Properties of Field Lines
1. Electric field lines indicate the direction of the electric field; the field points in the direction tangent to the field line at any point. 2. The lines are drawn so that the magnitude of the electric field, E, is proportional to the number of lines crossing unit area perpendicular to the lines. Closer the lines are together, the stronger the field. 3. Electric field lines start on positive charges and end on negative charges, and the number starting or ending is proportional to the magnitude of the charge. Also key to note that field lines never cross because it would not make sense for an electric field to have two directions at the same point.
Summary of Chapter 16 Short
1. Two kinds of electric charge-positive and negative 2. Charge is conserved 3. charge on electron: e = 1.602 * 10^-19 C 4. Conductors: electrons free to move 5. Insulators: nonconductors 6. Charge is quantized in units of e 7. Objects can be charged by conduction or induction. 8. Coulombs Law: F = k(Q1*Q2/r^2) in magnitudes 9. Electric field is force per unit charge: E = F/q 10. Electric field of a point charge: E = k* Q/r^2 11. Electric fields can be represented by electric field lines 12. Static electric fields inside conductors is zero, and the surface field will be perpendicular to their surface.
Parallelogram Method
A method used to find the resultant of two vectors in which you place the vectors at the same initial point, complete a parallelogram, and draw the diagonal. Where again F = F1 + F2.
Electric field is always perpendicular to the surface outside of a conductor
A related property of static electric fields and conductors is that the electric field is always perpendicular to the surface outside of a conductor. If there were a component of E parallel to the surface, it would exert a force on free electrons at the surface, causing the elctrons to move along the surface until they reached positions where no net force was exerted on them parallel to the surface-that is until the electric field was perpendicular to the surface.
Electroscope
An electroscope is a device that can be used for detecting charge. There will be a case where inside there are two movable metal leaves often made of gold foil connected to a metal knob on the outside. Sometimes only the leaf is movable.
Grounded or Earthed
Another way to induce a net charge on a metal object is to first connect it with a conducting wire to the ground or a conducting pipe leading into the ground. ⏚ means connected to the ground. The object is then said to be grounded or earthed. The earth because its so large and can conduct, easily accepts or gives up electrons, acting like a reservoir for charge. If a charged object that for example is negative, is brought close to a metal object, free electrons in the metal are repelled and many of them move down the wire into the Earth. This leaves the metal positively charged. If the wire is now cut, th emetal object will have a positive induced charge on it. If the wire is cut after the negative object is moved away, the electrons would all have moved from the ground back into the metal object and it would be neutral again.
Reasoning Behind Electric fields inside conductors
Any net charge on a conductor distributes itself on the surface. For a negatively charged conductor the negative charges repel one another and race to the surface to get as far from one another as possible. Another consequence is the following. Suppose a positive charge Q is surrounded by an isolated uncharged metal conductor whose shape is spherical. Because there can be no field within the metal, the lines leaving the central positive charge must end on negative charges on the inner surface of the metal. That is, the encircled charge + Q induces an equal amount of negative charge, -Q, on the inner surface of the spherical shell. Since the shell is neutral, a positive charge of the same magnitude, + Q, must exist on the outer surface of the shell. Thus, although no field exists in the metal itself, an electric field exists outside of it, as if the metal were not even there. (look at page 459)
Electric Field
Both gravitational and electrical force act over a distance: there is a force between two objects even when the objects are not touching. The idea of a force acting at a distance was difficult for earlier thinkers. But they used the idea of a field to help solve this. An electric field extends outward from every charge and permeates all of space. And if a second charge called Q2 is placed near the first charge, it feels a force exerted by the electric field that is there, say at point P. The electric field at point P is considered to interact directly with charge Q2 to produce the force on Q2.
Charge Separation in Nonconductors
Charge separation can also be done in nonconductors. If you bring a positively charged object close to a neutral nonconductor, almost no electrons can move about freely within the nonconductor. But they can move slightly within their own atoms and molecules. The negatively charged electrons, attracted to the external positive charge, tend to move in its direction within their molecules. So if you brought a positively charged object near a nonconductor, it would cause within the atoms the negative charges of the atoms to move slightly within their own atom towards the positive charge. Because the negative charges in the nonconductor are nearer to the external positive charge, the nonconductor as a whole is attracted to the external positive charge.
Elementary Charged
Charges produced by rubbing ordinary objects are typically around a microcoulomb (1 uc = 10^-6 C) or less. Objects that carry a positiv ehcarge have a deficit of electrons, whereas negatively charged objects have an excess of electrons. The charge on one electron has been determined to have. amagnitude of about 1.6022 * 10^-19, and is negative. This is the smallest observed charge in nature, and since its fundamental, its given the symbol e and is often referred to as the elementary charge: e = 1.6022 * 10^-19 C or about 1.6 * 10^-19 C.
Permitivity of Free Space
Constant k in Coulombs law is often written in terms of another constant, Epsilon 0 or ε0 is called the permittivity of free space. Its related to k by: k = 1/4*ε0* π. Thus this can be related to Coulombs law as: F = (1/4*ε0* π) (Q1*Q2/r^2) where ε0 = 1/(4* π*k) = 8.85 * 10^-12 C^2/N*m^2 Any of these forms can be used as they are all equivalent
Inverse Square Laws
Coulomb's Law looks a lot like the law of universal gravitation, F = G m1m2/r^2, which expresses the magnitude of the gravitational force a mass m1, exerts on a mass m2. Both are considered inverse square laws (where they are both proportional inversely to the square of r). Both also have a proportionality to a property of each object- mass for gravity, electric charge for electricity. And both act over a distance (that is there is no need for contact). A major difference between the two laws is that gravity is always an attractive force, whereas the electric force can be either attractive or repulsive. Electric charge comes in two types, positive and negative, whereas gravitational mass is only positive.
Central Region Electric Field Magnitude between two Plates
E = constant [Between two closely spaced, oppositely charged, flatt parallel plates]. In that central region, the electric field has the same magnitude at all points. The fringing of the field near the edges can often be ignored, particularly if the separation of the plates is small compared to their height and width.
Electroscope Examples
If a positively charged object is brought close to the knob, a separation of charge is induced: electrons are attracted up into the knob, and the leaves become positively charged. If instead, the knob is charged by conduction (touching), the whole apparatus acquires a net positive charge as the electrons from the metal knob will travel onto the positively charged object. This however does not tell you the sign of the charge, since negative charge will cause the leaves to separate just as much as an equal amount of positive charge. But either way the two leaves repel each other. An electroscope however, can be used to determine the sign of the charge if its first charged by conduction: for example, negatively. Then if a negative object is brought close, more electrons are induced to move down into the leaves and they separate further. If a positive charge is brought close instead, the electrons are induced to flow upward, so the leaves are less negative and their separatio nis reduced.
Superposition Principle in Electric Fields
If the electric field at a given point in space is due to more than one charge, the individual fields (E1, E2, etc.) due to each charge are added vectorially to get the total field at that point: E = E1 + E2 + ... This is demonstration again the superposition principle.
Coulombs law and magnitude
In coulombs law, it will give the magnitude of the electric force that either charge exerts on the other. The direction of the electric force is always along the line joining the two charges. If the two charges have the same sign, the force on either charge is directed away from the other and they repel . If the two charges have opposite signs, the force on one is directed toward the other (they attract). The force on charge exerts on the second is equal but opposite to that exerted by the second on the first as explained by Newtons Third Law.
Nuclei in solid materials
In solid materials the nuclei tend to remain close to fixed positions, whereas some of the electrons may move quite freely. When an object is neutral, it contains equal amounts of positive and negative charge. The charging of a solid object by rubbing can be explained by the transfer of electrons from one object to the other. This differs to liquids and gases where the nuclei or ions can m ove as well as electrons.
What is a conducting box used for?
It's an effective device for shielding delicate instruments and electronic circuits from unwanted electric fields. This is because the contents of the box won't be exposed directly to the electric fields, but the electric fields are still able to pass through the plates. That is why someone in a parked car is relatively safe from lightning storms where thy are surrounded by pure metal. The person is essentially inside a porous 'cage' which protects them from a strong electric discharge.
Polar Water Molecules
Normally when objects are charged by rubbing, they hold their charge only for al imited time and eventually return to the neutral state. This is because the charge 'leaks off' onto water molecules in the air. This is because water molecules are polar- meaning that even though they're neutral, their charge is not distributed uniformly. Thus the extra electrons on an object may 'leak off' into the air because they are attracted to the positive end of water molecules. A positively charged object on the other hand can be neutralized by transfer of loosely held electrons from water molecules in the air. This is why on dry days that static electricity is much more noticeable since the air contains fewer water molecules to allow leakage of charge. Whereas on humid or rainy days, its difficult to make any object hold a net charge for long.
Principle of superposition
Since coulombs law gives the force on a charge due to only one other charge, if several or many charges are present, the net force on any one of them will be the vector sum of the forces due to each of the others. This principle of superposition is based on experiment, and tells us that electric force vectors add like any other vector. For example, if you have a system of four charges, the net force on charge 1, is the sum of the forces exerted on charge 1 by charge 2,3, and 4. The magnitudes of these three forces however are still determined from Coulombs Law and then are added vectorially.
Electric Field Lines
Since electric fields are vectors, they are sometimes referred to as a vector field. We could indicate the electric field with arrows at various points in a given situation such as at A, B, and C like in the picture in the textbook, pg 457. The directions of EA, EB, and EC are the same as forces demonstrate earlier, but the magnitudes or arrow lengths are different since we divide F by q to get E. However, the relative lengths of EA, Eb, and Ec are the same as for the forces since we divide by the sam eq each time. To indicate the electric field in such a way at many points, however, would result in many arrows, which could quickly become cluttered and confusing. To avoid this, we use field lines as our technique. To visualize this electric field, we draw a series of lines to indicate the direction of the electric field at various points in space. These electric field lines or lines of force are drawn to indicate the direction of the force due to the given field on a positive test charge. The lines of force due to a single isolated positive and negative charge are shown in the picture. Where the lines point radially outward from the charge in the positive charged point, and point radially inward toward the charge in the negative one because thats the direction the force would be on a positive test charge in each case.
Why is E defined as F/q with q->0?
So E does not depend on the magnitude of the test charge q. Meaning that E describes only the effect of the charges creating the electric field at that point.
Charging by Induction
Suppose a positively chargd object is brought close to a neutral metal rod but does not touch it. Although the free electrons of the metal rod do not leave the rod, they still move within the metal toward the external positive charge, leaving a positive charge at the opposite end of the rod. A charge is said to have been induced at the two ends of the metal rod. No net charge has been created in the rod: charges have merely been separated. The net charge on the metal rod is still zero. However if the metal is separated into two pieces, we would have two charged objects: one positive and one negative. This is called charging by induction
Charging by Conduction
Suppose a positively charged metal object A is brought close to an uncharged metal object B. If the two touch, the free electrons in the neutral one are attracted to the positively charged object and some of those electrons will pass over to it. since object B, originally neutral, is now missing some of its negative electrons, it will have a net positive charge. This process is called charging by conduction.
Vector Addition Review
Suppose two vector forces, F1 and F2, act on an objec. They can be added using the tail to tip method or by the parallelogram method. These two methods are useful for understanding a given problem or getting a picture in your mind of what is going on. But for calculating the direction and magnitude of the resultant sum, its more precise to use the method of adding components.
What happens to electric fields direction when Q is positive vs negative?
The direction of the electric field toward a charge Q is given, and this dictates whether Q is positive or negative. The vice versa can also be determined. If a positive test charge is going towards Q, then the sign of the charge is negative (Q = -), but if it is going away from Q, then the sign of the charge is positive (Q= +). If given the charge of Q, you can assume whether the test charge is going towards or away from to help determine direction.
Measuring Electric Field at Any Point
The electric field at any point in space can be measured, based on the previous equation. For simple situations with one or several point charges, we can calculate E. For example, the electric field at a distance r from a single point charge Q would have the magnitude: E = F/q = (kqQ/r^2)/q or E = k(Q/r^2) or we can express it in terms of ε0 as: E = (1/(4*pi*ε0) (Q/r^2) These two/three equations are used for single point charges) E is independent of the test charge q, meaning E depends only on the charge Q which produces the field, and not on the value of the test charge q. So if we are given the electric field E at a given point in space, then we can calculate the force F on any charge q placed at that point by rearranging the equation to: F = q*E This is valid even if q is not small as long as q does not cause the charges creating E to move. If q is positive, F and E point in the same direction. If q is negative, F and E point in opposite directions.
Electric Dipoles
The electric field due to two equal charges of opposite sign leads to a combo known as an electric dipole. The eletric field lines are curved in this case and are directed from positive to the negative charge. The direction of the electric field at any point is tangent to the field line at that point as shown by thhe vector arrow E at point 1 on th ediagram. There is also the electric field lines for two equal positive charges and also electric field lines for unequal charges where they are -Q and +2Q. In the unequal one, twice as many lines leave +2Q as enter -Q which is showing the number of lines is proportional to magnitude of Q. There is also another example of a cross section where the field lines between two flat parallel plates carry equal but opposite charges. The electric field lines between the two parts start out as perpendicular to the surface of the metal plates, and go directly from one plate to the other as we would expect because a positive test charge placed between the plates would feel a strong repulsion from the positive plate and a strong attraction to the negative plate. The field lines betewen the two close plates are parallel and equall spaced in the central region, but as you go outwards, they begin to fringe outward near the edges.
Electric Field inside a Conductor
The electric field inside a conductor is zero in the static situation, that is, when the charges are at rest. If there were an electric field within a conductor, there would bea. force on the free electrons. The electrons would move until they reached positions where the electric field, and therefore the electric force on them, did become zero.
Electric Force between charged particles at rest
The electric force between charged particles at rest(referred to as electrostatic force or Coulomb force) is like all forces, a vector: it has both magnitude and direction. And when several forces act on an object (for example, call them F1, F2, etc.), the net force Fnet on the object is the vector sum of all the forces acting on it. This is again based on the principle of superposition for forces.
Coulomb's Law
The electric force one charged object exerts on a second charged object is directly proportional to the charge on each of them. That is, if the charge on either one of the objects is doubled, the force is doubled. And if the charge on both of the objects is doubled, the force increases to four times the original value. This was the case when the distance between the two charges remained the same. If the distance between them was allowed to increase, the force decreased with the square of the distance between them. That is, if the distance was doubled, the force fell to one fourth of its original value. Thus, Coulomb was able to conclude, the magnitude of the force F that one small charged object exerts on a second one is proportional to the product of the magnitude of the charge on one, Q1, times the magnitude of the charge on the other, Q2, and inversely proportional to the square of the distance 'r' between them. Thus as an equation, we can write Coulomb's Law as: F = k*(Q1*Q2/r^2) Where F is force, k is the proportionality constant, Q1 is charge 1, Q2 is charge 2, and r is the distance between the two charges. Its important to recognize that we are using the magnitude of the charges.
Free Electrons or Conduction Electrons
The electrons in an insulating material are bound very tightly to the nuclei. In a good metal conductor on the other hand some of the electrons are bound very loosely and can move about freely within the metal, but can't exactly just leave the metal. These are referred to as free electrons or conduction electrons. When a positively charged object is brought close too or touches a conductor, the free electrons in the conductor are attracted by this positively charged object and move quickly towards it. If a negatively charged object is brought close to the conductor, the free electrons in the conductor move swiftly away from it. In a semiconductor there are many fewer free electrons, and in an insulator, almost none.
Gravitational Field
The field ocncept can also be applied to gravity force. Thus we can say gravitational field exists for every object that has mass. One object attracts another by means of the gravitational field. The earth for example can be said to possess a gravitational field which is responsible for the gravitational force on objects. Gravitational force is defined as force per unit of mass. The magnitude of earths gravitational field at any point above the earths surface is thus G Me/r^2, where Me is the mass of earth, r is distance of the point from earths center, and G is teh gravitational constant. At earths surface, r is the radius of the earth and the gravitational field is equal to g, the acceleration due to gravity.
Point Charges
The previous equations of Coulombs law only really apply to objects whose size is much smaller than the distance between them. Ideally, its precise for point charges(spatial size negligible compared to other distances). For finite sized objects, its not always clear what value to use for r, particularly since the charge may not be distributed uniformly on the objects. If the two objects are spheres and the charge is known to be distributed uniformly on each, then r is the distance between two charges when they are at rest.
Simple model of an atom
The simple model of an atom shows it as having a tiny but massive, positively charged nucleus surrounded by one or more negatively charged electrons. The nucleus contains protons, which are positively charged, and neutrons which have no net electric charge. All protons and all electrons have exactly the same magnitude of electric charge; but their signs are opposite. Hence neutral atoms having no net charge, contain equal numbers of protons and electrons.
Semiconductors
There are a few materials which are like an intermediate category where they are known as semiconductors and are alright at transferring that charge. Examples of these are silicon and germanium.
Electrostatic Force
Thus, Coulombs Law describes the force between two charges when they are at rest. Additional forces come into play when charges are in motion. The study of this is called electrostatics, and Coulombs Law gives the electrostatic force
Two types of electric charge
Two types of electric charge are referred to as positive and negative.
Number of Lines on Electric Field Diagram and Magnitude of Charge
We can draw the lines of these electric fields so that the number of lines starting on a positive charge or ending on a negative charge is proportional to the magnitude of the charge. The nearer the charge, where the electric field is greater, the lines are closer together. This is a general property of electric field lines: the closer together the lines are, the stronger the electric field in that region. In fact, field lines can be drawn so that the number of lines crossing unit area perpendicular to E is proportional to the magnitude of the electric field.
Investigation of the Electric Field with Test Charge
We can investigate the electric field surrounding a charge or group of charges by measuring the force on a small positive test charge which is at rest. By a test charge, it means a charge so small that the force it exerts does not significantly affect the charges that create the field. If a tiny positive test charge q is placed at various locations in the vicinity of a single positive charge Q, so in this example points A, B, and C, the force exerted on q varies. (IN text book picture) The force at B is less than A because B's distance from Q is greater (Coulombs Law). And the force at C is smaller still for the same reason. In each case, the force on q is directed radially away from Q. The electric field is defined in terms of the force on such a positive test charge. In particular the electric field, E, at any point in space is defined as the force F exerted on a tiny positive test charge placed at that point divided by the magnitude of the test charge q: E = F/q E is defined as the limit of F/q as q is taken smaller and smaller, approaching zero. F is again the force exerted on a tiny positive test charge, and q is the magnitude of the test charge. So q is so tiny that it exerts essentially no force on other charges which created the field. And from this equation, we see that the electric field at any point in space is a vector whose direction is the direction of the force on a tiny positive test charge at that point, and whose magnitude is the force unit per charge. Thus E has SI units of newtons per coulomb (N/C).
Law of Conservation of electric charge
When a certain amount of charge is produced on one object, an equal amount of the opposite type of charge is produced on another object. Thus the net change in the amount of charge produced is zero. Even if the charges are separated, the sum of the two is still zero. This is the law of conservation of electric charge: which is that the net amount of electric charge produced in any process is zero. No net electric charge can be created or destroeyd.
Ion
When an atom may lose one or more of its electrons, or when they may gain electrons, they can have a net positive or net negative charge. This is an ion.
Adding Electric Forces: Principle of Superposition
When dealing with several charges, is helpful to use double subscripts on each of the forces involved. The first subscript referring to the particle on which the force acts, the second referring to the particle that exerts the force. For example, if we have 3 charges, F31 means the force exerted on particle 3 by particle 1. Its also important to remember when solving these problems, drawing diagrams can be very helpful, particularly free body diagrams for each object to show all the forces acting on that object. In applying Coulombs Law, we can deal with charge magnitudes only to get the magnitude of each force. Then determine separately the direction of the force physically (along the line joining the two particles: do the charges repel or attract), and then lastly, show the force on the diagram. then lastly, add all the forces on one object together as vectors to obtain the net force on that object.
Calculating the Direction and Magnitude of the Vectors in Coulombs Law.
When you have two vectors, and you are trying to determine the direction and magnitude of the resultant sum, it is always a good strategy to break the vectors into their components. The vectors can be broken down into their x and y components, forming triangles, and the x and y components of their vectors make up two of the sides of the triangle. And then also one of the angles in each of their triangles besides the right angle will be Theta (θ). For example, two vectors F1 and F2. When resolved into their x and y components, will produce: F1x and F1y, as well as θ1 in between the normal vector line and F1x. As well as F2x, F2y, and θ2 in between the normal vector line and F2x. Using trigonnometric functions, we can determine: F1x = F1 * cosθ1 F1y = F1 * sinθ1 F2x = F2 * cosθ2 F2y = -F2 * sinθ2 Any of them could be positive or negative depending on where the vectors lie on a graph. Then we can add up the x and y components separately to obtain the components of the resultant Force F: Fx = F1x + F2x = F1cosθ1 + F2cosθ2 Fy = F1y + F2y = F1sinθ1 - F2sinθ2 Again, it could end up as subtraction or addition, all depends on the direction of each of the vectors in relation to one another. Thus the magnitude of the resultant or net force F is: F = Square Root of (F^2x + F^2y) Due to Pythagorean Theorem And the directio nof F is specified by the angle θ that F makes with the x axis, which is given by tanθ = Fy/Fx
Do the properties listed above apply to only conductors?
Yes these properties apply only to conductors. AS inside a nonconductor, which doesn't have free electrons, a static electric field can exist. Also, the electric field outside a nonconductor does not necessarily make an angle of 90 degrees to the surface.
Tail to Tip Method
a type of vector addition in which the first vector is drawn to scale with the tip of the arrow being the starting point for the next vector and where the resultant is drawn from the origin of the first vector to the arrow head of the last. The total or net force will end up being F = F1 + F2 and represent the line that connects the tail of one of the vectors to the tip of the other vector.
Electrons Quantized
e is considered a positive number, so the charge on electrons are actually -e. Where as the charge of the proton is actually +e. Since an object can't gain or lose a fraction of an electron, the net charge on any object must be an integral multiple of this charge. Electric charge is thus said to be quantized or existing only in discrete amounts : 1e, 2e, 3e, etc. Since e is so small, however, we normally don't notice this discreteness in macroscopic charges, which thus seem continuous.
List some properties of conductors:
the electric field inside a conductor is zero in the static situation when charges are at rest and electric fields are always perpendicular to the surface outside of a conductor.
