Electrostatic 1 Force An electromagnetic force exists between two objects provided both bodies have an electrical charge. In a neutral ground-state atom, the electromagnetic forces exerted by the negatively-charged electrons on some external charged body are canceled by the forces exerted by the equal number of positively-charged protons on that object. The negative electrons furthest from the attractive force of the atom's positive proton-bearing nucleus are capable of being removed from one atom and added to another. Like-charged particles will repel each other whereas oppositely-charged particles attract one another. A net negative charge is due to an excess of electrons while a net positive charge is due to a loss of electrons relative to the number of protons present in the object. The Electroscope The detection of electromagnetic forces requires either the addition or removal of negative electrons. If both bodies contain an excess of electrons (negatively-charged) or an excess of protons (positively-charged) a repulsive electric force exists between the objects. Electrons from atoms of fur or wool tend to migrate to rubber, giving the material a net negative charge. Glass tends to transfer electrons from its atoms to silk, giving the glass a net positive charge. An electroscope consists of a metal rod to which is attached a free-moving conductor or leaf. The rod passes through an insulator into a metal enclosure which surrounds the leaf. An object (rod) is typically touched to (conductive-charging) or held nearby (inductive-charging) the knob causing the movement of electrons into leaves or from the leaves to the knob. An excess or deficiency of electrons in the leaves will result in a repulsive effect between like charges causing the leaves/needle to repel each other. Note that the movement of the leaves does not indicate the type of charge on the electroscope. Charging by conduction occurs whenever a body acquires a net charge of equal magnitude and sign as the charging body. To charge an electroscope by conduction, a positively or negatively charged object makes physical contact with the knob of the electroscope. If a positively-charged glass rod is placed nearby the scope, negative electrons are drawn from the electroscope leaves to the knob. As the electrons from the knob and leaves of the electroscope are transferred into the rod, the entire scope is left with a net positive charge. Charging by conduction means that the charging body contacts the electroscope's knob. With an ebonite rod rubbed with fur, electrons transfer from the negative rod to the knob resulting in a net negatively-charged scope. Electroscope charged by conduction where electrons are transferred from the leaves/knob of the to a positively-charged glass rod. A deficit of negative electrons on the leaves yields a residual net positive charge on the electroscope. When a negatively-charged rubber rod is brought close to the knob of the uncharged electroscope rod, the free electrons of the atoms of the knob are repelled downward from the knob to the leaves. The leaves are electron-saturated [negativelycharged] such that they repel each (outward deflection of the leaves). Withdrawing the rubber rod causes the electrons in the leaves to be attracted upward to the knob, thereby re-creating the initial neutral conditions of the scope (leaf collapses). If the rubber rod touches the knob, electrons are transferred from the rod to the knob/leaves. The scope would become net negatively-charged by the addition of the
2006-2007 Ronald J. Maniglia
2 electrons. Electroscope charged by conduction where electrons are transferred to the leaves/knob from a negatively-charged rubber rod. A surplus of negative electrons on the leaves yields a residual net negative charge on the electroscope. When an electroscope is charged by induction, it acquires a net residual charge of equal magnitude but opposite sign to the charging body. Charging by induction means that the charging rod is brought close to the electroscope's knob. If the electroscope is not grounded, it will remain neutral but temporarily polarized while the charging rod is in the immediate vicinity. A positive rod will induce the electrons in the scope to migrate to the knob, leaving the leaves of the scope 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 into the knob if the charging rod is positive. Upon removing the grounding wire, the electroscope is left with a residual charge that is opposite to that of the charging rod. There is no direct transfer of electrons from the scope to the glass rod. A deflection of the leaf is observed due to the like charges existing on both the rod and the leaf. A ground serves as a pathway for the movement of electrons either to or from the scope. When detecting an "unknown charge", if a rod of equal sign were to be brought near a scope, the leaf would further deflect indicating the presence of a like charge. The electroscope leaf would collapse if an oppositely-charged rod were introduced in the area of the knob. A charged scope can be uncharged or neutralized by grounding. Grounding is the process of removing the excess charge on an object by means of the transfer of electrons between it and another object. Nature of Charge Charge is quantized in that it appears only as integral multiples of a fundamental indivisible unit, the electron charge equal to 1.6 x 10-19 Coulombs. One coulomb (C) of charge represents an excess or deficit of 6.24 x 10 18 electrons. A positively-charged object has a deficit of electrons. Each electron lost gives the object a charge of +1.6 x 10-19 coulombs. Positive charge can be created by rubbing a glass rod with silk. According to the law of conservation of charge, the total charge in an isolated system remains constant. The system of the glass rod and silk maintains a net charge of zero. A negatively-charged object acquires electrons. Each electron gained gives the particle a charge of -1.6 x 10-19 coulombs. Negative charge can be created by rubbing a rubber rod with fur. The transfer of electrons from the fur leaves a positive charge, while the rod becomes negatively-charged by the gain of electrons. Compared to an identical neutral body, negatively-charged objects have more mass owing to the mass (9.11 x 10-31kg) of each extra electron.
Coulomb's Law According to Coulomb's law concerning two charged bodies q 1 and q2 of small dimensions relative to the distance between the objects, the magnitude of the electrostatic force Fe is directly proportional to the product of the absolute value of the magnitude of their charges and inversely proportional to the square of the linear distance separating the charges.
2006-2007 Ronald J. Maniglia
3
The direction of the force is in keeping with the principle that like charges repel and unlike charges attract each other. If an appropriate proportionality constant (k) is selected, the equation becomes:
F
=k e
q1q2 where 2 r
k =9x10 9 Nm2 C2
According to the superposition principle, the force acting on a charge located at some point in a region that contains other charges is the equal to the vector sum of the forces due to all the charges. Electric Field An electric field E at a point is defined as the force that an infinitesimally small unit positive charge would experience if positioned at that point (The assumption is being made that the test charge does not affect the distribution of any charges in the vicinity). Mathematically, since E=F/q then this vector relationship can be stated as follows:
E=
kq r
2
The direction of the electric field at a given point is in keeping with the direction of the force experienced by a unit positive charge placed at that point. Electric Field Lines Michael Faraday introduced the concept of electric field lines as a visual representation of the magnitude and direction of the electric field in a region of space with the following restrictions: The lines are always directed from a positive charged particle towards a negative particle such that the tangent to any such field line at a point indicates the direction of the electric field at that point; The number of lines emerging or terminating on a charge is proportional to the magnitude of the charge; and, Every field line must originate in a radial manner at a positive charge and terminate at a negative charge in such a manner that they do not intersect in a region free of charge [an intersection would denote both a state of positiveness and negativeness assigned to the same point or mathematically, a relation rather than a function would describe the field].
Gauss' Law Whereas Coulomb's law applies to static charges, Gauss' law can account for the interactions experienced by a moving charge. To simplify the derivation of the formula, it is valid to assume that the number of electric field lines N is proportional to the magnitude of the charge Q. Consequently, if a spherical region (surface) is hypothesized at a fixed radial distance r from a charge Q,
2006-2007 Ronald J. Maniglia
4 then the density of the field lines crossing perpendicular (normal) the surface per unit area (electric flux ) would be given by N/4r2. Since the number of lines N is proportional to Q which in turn is proportional to E, then as r approaches zero, the density would be directly related to the magnitude of the field, regardless of the shape of the surface. The electric field outside a sphere must be the same as the field from a point charge with a net charge of Q. The implication of Gauss' law is that at equilibrium, the excess charge lies only at the surface of the conductor. While the electric field is zero within the solid part of the conductor, the field is perpendicular to the surface. The electric flux is a measure of the number of electric field lines passing through an area. To calculate the flux through a particular surface, the surface area is multiplied by the component of the electric field perpendicular to the surface. If the electric field is parallel to the surface, no field lines pass through the surface and the flux will be zero. The maximum flux occurs when the field is perpendicular to the surface. where is the angle between the Q electric field and Normal to the surface. (cos )
0
According to Gauss' law, the sum of the electric flux through a surface is equal to the charge enclosed by a surface divided by the permittivity of free space constant (o) which equals from Coulomb's law 1/(4k) or 8.85x10-12Nm2C2. Gauss' law states that the net number of lines crossing normal to any closed surface is directly proportional to the magnitude of the net charge contained within that surface. The surface integral of the normal component (N) of the electric field over any closed surface equals 4k times the net charge contained insider the surface. Note that the surface integral is proportional to the enclosed charge q, regardless of the shape or size of the surface or the location of the charge. If q were negative, E would be directed radially inward rather than radially outward as when q is positive. An area integral of a vector function E can be defined as the integral on a surface of the scalar product of E with area element dA. The direction of the area element is defined to be perpendicular to the area at that point on the surface. For a Gaussian surface at equilibrium: The net electric charge of a conductor resides entirely on its surface. (The mutual repulsion of like charges from Coulomb's Law demands that the charges be as far apart as possible, hence on the surface of the conductor.) The electric field inside the conductor is zero. (Any net electric field in the conductor would cause charge to move since it is abundant and mobile which violates the condition of equilibrium.) The external electric field at the surface of the conductor is perpendicular to that surface. (If there were a field component parallel to the surface, it would cause mobile charge to move along the surface negating equilibrium.)
Faraday's Ice Pail Experiment In Faraday's ice pail experiment, a charge Q is placed at the center of metallic a shell. The electric field due to this charge penetrates the surface. The resulting electric field within the metal surface opposes that of the field generated by the charge Q such that no field exists within the metal surface.
2006-2007 Ronald J. Maniglia
5 An electroscope will indicate an induced charge on the container. The charge on the inside of the container is opposite to the charged sphere, while the charge on the outside is the same as the sphere. Moving the sphere inside the container has no effect on the leaves of the electroscope. The excess charge therefore resides on the outer surface equal in magnitude to that of the charge Q. If the charge were to contact the inner surface, the excess charge on the cavity wall would then neutralize the charge Q, leaving the inner surface completely uncharged, effectively (conducting) transferring the entire charge Q to the outer surface. If the shell were grounded, no charge would remain on the outer surface (although a charge would remain on the inner surface). In this case, no field exists outside the shell due to the charge Q. When Faraday suspended a positively charged metal ball into the ice pail, the leaves of the electroscope diverged 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. If the metal ball was allowed to contact the inside surface of the ice pail, the leaves diverged on the scope. When the ball was completely removed from the inside of the ice pail, the leaves remained diverged on the scope. The metal ball was no longer charged because the inner surface of the pail had enough charge to neutralize the ball. 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 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 appears on the outer surface of the container. Electric fields can be shielded by surrounding the surface 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. Electric Potential Electric potential defined throughout an electric field in a region of threedimensional space describes the electric potential energy that will be possessed by a unit positive test charge when placed at various locations in the field. The electric potential at a particular point is the sum of the potentials due to all the source charges. The potential difference V between two points is the work performed against the electrostatic field in moving a unit positive test charge from one point to another. Zero potential is considered to exist at a point infinitely distant from any unbalanced charges. As the electrostatic field is a conservative field, the electric potential difference is a scalar quantity. The work performed is independent of the path taken by the charge between any two points. If a negative charge is moved through a potential difference V, then the potential energy will decrease as the charge moves from a lower to a higher potential. [A negative charge seeks a higher potential]. Likewise, a positive charge would tend to move towards a point of lower potential, gaining kinetic energy equal to the loss in potential ("voltage drop"). After positive work done on the system separates oppositely-charged particles and increases its potential energy, positive work done by the system pulls the particles together and decreases its potential energy again. Whether on or by a system, the work done equals a force applied through a displacement. In terms of work done on the
2006-2007 Ronald J. Maniglia
Given : E = k V A - V B = kq( 1
q r
2
- ) r A work (force in rB system, separating two charges with opposite signs takes positive Assuming r is at infinity,then V B = 0 direction of displacement) and increases their potentialBenergy.
1
6 the
W AB = q(V B - V A )
V point =
kq r
It is possible that E = 0 and V 0 and E 0 and V = 0 at the same point. The field is directed away from positive point charges, and toward negative point charges. By the principle of superposition, the net field due to two charges equals the vector sum of their individual fields. The electric potential midway between two equal but opposite charges - an electric dipole - equals zero. Parallel Charged Plates & Equipotential Surfaces Consider two charged parallel conducting plates separated by a distance (d). There is a significant relationship between the electric field and the electric potential. Recall that the work performed by an electric field in moving a positive charge q from points A to B is given as W=q(V B-VA). Work is defined as the force F exerted over a distance (d), which in this case is the separation between two parallel conductors. Given W=Fd=q(VB-VA) then F=qE such that VB-VA=Ed. The electric field between two parallel conductors separated by a distance d is equal to
E
VBA d
An electron volt (eV) is defined as the increase in the potential energy of a charge of e coulombs when "raised" through a potential difference of 1 volt. [1eV=1.6 x 10-19J]. The electric potential between the plates increases in the opposite direction to the field E. Since it takes positive work to move a test charge against the field, the potential energy of the charge increases as it moves from the negative to the positive plate. The field is always directed from regions of higher potential to regions of lower potential. Returning to Gauss's law, in any region where E=0 at all points as in the interior of a conductor with charges at rest, the potential difference between any two points is zero. In other words, all points in the region have the same potential. The interior of a charged conductor is therefore an equipotential surface. An equipotential surface must be perpendicular to the electric field at any given point. The equipotential surface(s) near an isolated point charge (or any other conducting surface) are always perpendicular to the electric field lines. Any motion of a charged particle from one place to another along such a surface must be in a direction perpendicular to the force exerted by the field. Given that work equals the component of the force parallel to the motion multiplied by the displacement, with no parallel component, no work occurs in moving a charge along an equipotential surface. [Note that every metallic surface is an equipotential surface]. As a charge is approached from infinity, [the electric field increasing as r decreases] the separation between equipotential surfaces decreases provided the difference in potential between neighboring surfaces is kept constant.
2006-2007 Ronald J. Maniglia
7 Capacitance A capacitor consists of two closely-spaced conductors carrying equal and opposite charges such that the capacitance (C) is defined as the ratio of the magnitude of the charge (Q) on either conductor to that of the potential difference (V) between the conductors. By accumulating charge, the capacitor also possesses a stored potential energy (U).
C=
Q V
1 1 U = QV = CV 2 2 2
For a given parallel-plate capacitor, the capacitance is directly proportional to the area of the plates (A) and inversely proportional to their separation (d). Since the degree to which a capacitor can store charge is limited by the dielectric strength of the surrounding medium, a non-conducting or insulating material is usually inserted between the plates. In response to the external electric field generated by the capacitor's plates, electrons within the dielectric material reposition themselves, such that the constituent atoms form dipoles. The induced internal electric field Ep opposes the existing field between the plates Eo, thereby reducing the overall electric field intensity. Any reduction in the net electric field will cause a proportional drop in the potential difference between the plates thereby increasing the capacitance. The dielectric constant K for a given non-conducting material is defined as the ratio between the capacitance with the insertion of a dielectric to that of the capacitance in a vacuum [K=1].
K=
C
Co where is the relative permittivity = 8.85x10-12 C 2 Nm 2
C = K
A d
C1 = C Q1 = CV1 V1 = U1 = C1V12
C2 = Q2 = V2 = U2 =
[K]C1 Q1 /[K] U1/[K]
C3 = Q3 = V3 = U3 =
C2 [K]Q2 U1[K]
The dielectric increases the ability of the capacitor to store charge (capacitance) by reducing the potential difference between the plates. Note that the emf source is initially disconnected upon insertion of the dielectric to avoid damage to the capacitor. When reintroduced to the capacitor, the potential difference returns to the original state thus allowing the accumulation of additional charge owing to the increased capacitance.
2006-2007 Ronald J. Maniglia
8 Electromagnetic Phenomenon Some basic magnetic phenomenon is summarized concerning the properties of charged particles. Magnets are dipoles whose properties are due to the arrangement of electrons in their atoms. Magnetic monopoles are not known to exist in nature. Magnetic Field A magnetic field () exists in a region of space if a unit positive charge (q) moving through the region with a velocity (v) at an angle with respect to the field experiences a magnetic force (F). The magnetic field intensity is measured in units known as Tesla.
F = qv sin
To find the direction of the magnetic force on a positive charge using the right-hand rule, start by pointing your first finger of your right hand in the direction of the velocity (v). Point your second finger in the direction of the magnetic field (B) such that your thumb points in the direction of the force (F).
Lorentz Force Whenever a charged particle (q) travelling with velocity (v) simultaneously comes under the influence of a magnetic () and an electrical (E) field, it will experience a force equal to the sum of the magnetic and electrical forces acting upon the particle.
F Lorentz = q(E + v sin )
Magnetic Flux Gauss' law for magnetic fields states that the total magnetic flux penetrating a closed surface must equal zero. The magnetic field or flux density () is equal to the number of magnetic field lines or magnetic flux () per a given area (A). The equation employs the cosine rather than the sine of the angle whenever a line drawn normal to the surface area (A), makes an angle () with the direction of the magnetic field. Gauss' law for magnetic fields is one of four of Maxwell's equations for magnetic phenomenon.
=
A sin
Motion of a Charged Particle in a Magnetic Field Whenever a charged particle q moves with a velocity v perpendicular to a uniform magnetic field B, the generated magnetic force F will act as a centripetal force causing the particle to move in a circular path of radius r. If the initial direction of the particle's velocity is not perpendicular to the field, the path will be helical along the magnetic field lines.
r=
mv q sin
2006-2007 Ronald J. Maniglia
9 Mass Spectrometers Mass spectrometers are used to determine the masses of atoms or molecules, or their relative abundance in a sample. After vaporizing the sample, an electron is removed through ionization to yield particles with a net positive charge. The ionized particles are accelerated across a potential difference (V) between two charged plates kept constant with respect to the resulting electric field (B). Since the force exerted on various particles by the electric field does not change, their accelerations depend only on their masses. The particles move in a circular path as they pass through an entry port into a uniform magnetic field perpendicular to their velocity. The magnetic field is kept constant so that the radius of the path is a function of the particle's speed and mass. Each particle traverses a circular path before striking the recording plate. The radius of the path is one-half the distance from the entry port.
qr 2B 2 m 2V
2006-2007 Ronald J. Maniglia