# Where k = 1. The electric field produced by a point charge is given by

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1 Ch 21 review: 1. Electric charge: Electric charge is a property of a matter. There are two kinds of charges, positive and negative. Charges of the same sign repel each other. Charges of opposite sign attract. Charge is conserved. 2. Coulomb s law: For charges q 1 and q 2 separated by a distance r, the magnitude of the electric force on either charge is given by F = 1 q 1 q 2 4πε r 2 Where k = 1 4πε = N m 2 /C 2 ; and ε = C 2 /N m 2. The force on each charge is along the line joining the two charges repulsive if the two charges have the sign, attractive if they have opposite signs. 3. Electric field: A charged body produces an electric field in the space around it. The electric field is force per unit charge exerted on a test charge at any point. E = F q The electric field produced by a point charge is given by E = 1 q 4πε r 2 r The electric field of any combination of charges is the vector sum of the fields produced by the individual charges. 4. Electric field lines: Electric field lines provide a graphical representation of electric fields. At any point on a field line, the tangent to the line is in the direction of the field at that point. The number of lines per unit area is proportional to the magnitude of the field at the point. 5. Electric dipole: An electric dipole is a pair of electric charges of equal magnitude q but opposite sign, separated by a distance d. The electric dipole moment p has magnitude p = qd. The direction of p is from negative toward positive charge. In an electric field E, an electric dipole experiences a torque τ = p E τ = pe sin φ 6. How to set up an integral: For a given charge distribution, the total electric field at a point is the vector sum of the fields at this point due to each point charge in the charge distribution. If the charges are continuously distributed along a line, over a surface, or through a volume, i.e. the charges cannot be considered as discrete point charges, it requires to integrate over the charge distribution to calculate the total electric field. Here is the general guideline how hot set up an integral for this type of problems. a. Make a drawing showing the distribution of the charges and your choice of coordinate system. 1

2 b. Divide the charge distribution into infinitesimal segments. Try to use symmetry and do it in a way such that one of the electric field components can be cancelled. c. Calculate the electric field due to an infinitesimal segment and express the electric field in terms of variables so that the expression can be generalize to all segments. d. Construct an integral based on this expression and integrate over the whole charge distribution. e. Calculate the integral to find the total electric field. Ch 22 review: 1. Electric flux: Electric flux is a measure of the flow of electric field through a surface. It is equal to the product of an area element and the perpendicular component of E, integrated over a surface. Φ E = E cos φ da = E da = E da 2. Gauss s law: Gauss s law states that the total electric flux through a closed surface, which can be written as the surface integral of the component of E normal to the surface, equals a constant times the total charge Qencl enclosed by the surface. Gauss s law is logically equivalent to Coulomb s law, but its use greatly simplifies problems with a high degree of symmetry. Φ E = E da = Q encl ε 3. Charges on a conductor: when excess charge is placed on a conductor, the charge resides entirely on the surface and electric field is zero everywhere in the conductor. Ch 23 review: 1. Electric potential energy: The electric force caused by any collection of charges at rest is a conservative force. The work W done by the electric force on a charged particle moving in an electric field can be represented by the change in a potential-energy function U. The electric potential energy for two point charges q and q depends on their separation r. The electric potential energy for a charge q in the presence of a collection of charges q1, q2, q3 depends on the distance from q to each of these other charges. W U U U a b a b 1 qq two point charges 4 r 2

3 U q q q q 4 r r r q qi i r i q in presence of other point charges 2. Electric potential: Potential, denoted by V, is potential energy per unit charge. The potential difference between two points equals the amount of work that would be required to move a unit positive test charge between those points. The potential V due to a quantity of charge can be calculated by summing (if the charge is a collection of point charges) or by integrating (if the charge is a distribution). The potential can also be calculated if the electric field is known. U 1 q V q 4 r U 1 qi V q 4 i r 1 dq V 4 r b i due to a point charge V V Ed l = E cos dl a b a due to a collection of point charges due to a charge distribution b a 3. Equipotential surfaces: An equipotential surface is a surface on which the potential has the same value at every point. At a point where a field line crosses an equipotential surface, the two are perpendicular. When all charges are at rest, the surface of a conductor is always an equipotential surface and all points in the interior of a conductor are at the same potential. When a cavity within a conductor contains no charge, the entire cavity is an equipotential region and there is no surface charge anywhere on the surface of the cavity. 4. Finding electric field from electric potential: If the potential V is known as a function of the coordinates x, y, and z, the components of electric field E at any point are given by partial derivatives of V. V x y V E E Ez V x y z ˆ V ˆ V ˆ V E = i j k vector form x y z 3

4 Ch 24 review: 1. Capacitors and capacitance: A capacitor is any pair of conductors separated by an insulating materials. The capacitance is Q C V A For a parallel-plate capacitor: C = ε d r For a spherical capacitor: C = 4πε a r b r b r a For a cylindrical capacitor: C = 2πε L ln ( r b ); C/L = 2πε ln ( r b ra 2. Capacitors in series and parallel: For capacitors in series, each capacitor has the same amount of charge. The equivalent capacitance C eq satisfies 1 C eq = 1 C C C 3 + For capacitors in parallel, each capacitor has the same potential difference. The equivalent capacitance is ab ra ) C eq = C 1 + C 2 + C Energy in a capacitor: The energy U required to charge a capacitor C to a potential difference V and a charge Q is equal to the energy stored in the capacitor. The energy stored in a capacitor is 2 Q 1 2 U 1 2CV 2QV 2C The energy density of an electric field is u 1 2 E 2 4. Dielectric: When a dielectric material is inserted between the plates, an induced charge of the opposite sign appears on each surface of the dielectric. The induced charge is a result of redistribution of positive and negative charge within the dielectric materials, a phenomenon called polarization. The permittivity is defined as ε = Kε. The capacitance with the dielectric is A A C KC K (parallel-plate capacitor filled with dielectric) d d Ch 25 review: 1. Current and current density: Current is the amount of charge flowing through a specified area. 4

5 I = dq dt = n q v da The current density is current per unit cross-sectional area: J = nqv d 2. Resistivity: The resistivity ρ of a material is the ratio of the magnitude of electric field and current density. ρ = E J Resistivity usually increases with temperature; for small temperature changes this variation is represented approximately by ρ(t) = ρ [1 + α(t T )] 3. Resistors: The potential difference V across a sample of material that obeys Ohm s law is proportional to the current through the sample. V = IR And the resistance R is R = ρl A 4. Circuits and emf: A complete circuit with a continuous current-carrying path and a source of electromotive force can carry a steady current. The terminal potential difference is V ab = E Ir 5. Energy and power in circuits: A circuit element with a potential difference and a current puts energy into a circuit if the correction direction is from lower to higher potential in the device, and it takes energy out of the circuit if the current is opposite. The power equals the product of the potential difference and the current. P = V ab I (general circuit element) P = V ab I = I 2 R = V ab 2 (power into a resistor) R Ch 26 review: 1. Resistors in series and parallel: When several resistors are connected in series, the current through all resistors is the same, the equivalent resistance is R eq = R 1 + R 2 + R 3 + When several resistors are connected in parallel, the potential difference across all resistors is the same, the equivalent resistance satisfies 1 R eq = 1 R R R 3 + 5

6 2. Kirchhoff s rules: Kirchhoff s junction rule states that the algebraic sum of the currents into any junction must be zero. Kirchhoff s loop rule states that the algebraic sum of potential differences around any loop must be zero. I = (Junction rule) V = (Loop rule) 3. R-C circuits: When a capacitor is charged by a battery in series with a resistor or discharges through a resistor, the current and capacitor charge are not constant. During the charging: q = CE (1 e RC) t = Q f (1 e RC) t During the discharging: i = dq dt = E R e t/rc = I e t/rc i = dq dt = Q q = Q e t/rc t RC e RC = I e t/rc The time constant τ = RC is a measure of how quickly the capacitor charges or discharges. Ch 27 review: 1. Magnets: Permanent magnet has two poles: north and south. North and south poles always appear in pairs. Opposite poles attract and like poles repel. 2. Magnetic forces: A particle with charge q moving with velocity v in a magnetic field B experiences a force F that is perpendicular to both v and B. F = qv B 3. Magnetic field lines and flux: A magnetic field can be represented graphically by magnetic field lines. At each point a magnetic field line is tangent to the direction of B at that point. The field magnitude is large where field lines are close together, and vice versa. Magnetic flux through an area is defined in analogous way to electric flux. The SI unite of magnetic flux is the weber (1 Wb = 1 T m 2 ). The net magnetic flux through any closed surface is zero (Gauss s law for magnetism). Magnetic field lines always close on themselves. Φ B = B da = B cos φ da = B da B da = (closed surface) 4. Motion in a magnetic field: The magnetic force is always perpendicular to v, a particle moving under the action of a magnetic field alone moves with constant speed. In a uniform field, a particle with initial velocity perpendicular to the field moves in a circle with radius R given by 6

7 R = mv q B The time to complete a full circle is given by T = 2πR The cyclotron frequency is f = 1 T = B q 2πm v = 2πm q B. Crossed electric and magnetic fields can be used as a velocity selector. 6. Magnetic force on a conductor: A straight segment of a conductor carrying current I in a uniform magnetic field B experiences a force given by F = Il B A similar relationship give the force df on an infinitesimal current carrying segment dl: df = Idl B 7. Magnetic torque: A current loop with area A and current I in a uniform magnetic field B experiences no net magnetic force, but does experience a magnetic torque. τ = IBA sin φ Ch 28 review: Define the magnetic dipole moment μ = IA, then τ = μ B U = μ B = μb cos φ 1. Magnetic field of a moving charge: The magnetic field B created by a charge q moving with velocity v is given by B = μ qv r 4π r 2 2. Magnetic field of a current-carrying conductor: The law of Biot and Savart gives the magnetic field db created by an element dl of a conductor carrying current I. db = μ Idl r 4π r 2 3. Magnetic field of a long, straight, current-carrying conductor: The magnetic field B at a distance r from a long, straight conductor carrying a current I is given by B = μ I 2πr The magnetic field lines are circles coaxial with the wire, with directions given by the right-hand rule. 4. Magnetic force between current-carrying conductors: Two long, parallel, currentcarrying conductors attract if the currents are in the same direction and repel if the currents are in opposite direction. The magnetic force per unit length between the conductors is given by 7

8 F L = μ II 2πr 5. Magnetic field of a current loop: The magnetic field along the axis of a circular conducting loop of radius a carrying current I is given by Ch 29 review: B x = μ Ia 2 2(x 2 +a 2 ) 3/2 (a circular loop) Bx = μ NI 2a 1. Faraday s law: Faraday s law states that the induced emf in a closed loop equals the negative of the time rate of change of magnetic flux through the loop. E = dφ B dt 2. Lenz s law: Lenz s law states that an induced current of emf always tends to oppose or cancel out the change that caused it. It is often easier to use Lenz s law to determine the direction of the induced current or emf. 3. Motional emf: If a conductor moves in a magnetic field, a motional emf is induced. For a conductor with length L moves in a uniform B field, L and v are both perpendicular to B and to each other, then E = vbl 4. Induced electric fields: When an emf is induced by a changing magnetic flux through a stationary conductor, there is an induced electric field of nonelectrostatic origin. Ch 32 review: E dl = dφ B dt 1. Maxwell s equations and electromagnetic waves: Maxwell s equations predict the existence of electromagnetic waves that propagate in vacuum at the speed of light c. Electromagnetic waves are transverse; the E and B fields are perpendicular to the direction of propagation and to each other. Faraday s law and Ampere s law give relationships between the magnitudes of E and B. In a plane wave, E and B are uniform over any plane perpendicular to the propagation direction. E = cb; c = 1 ε μ 2. Sinusoidal electromagnetic waves: A sinusoidal plane electromagentical wave traveling in vacuum in the +x-direction can be described by E (x, t) = j E max cos(kx ωt); B (x, t) = k B max cos(kx ωt) 3. Energy in electromagnetic waves: The energy flow rate (power per unit area) in an electromagnetic wave in vacuum is given by the Poyting vector. The magnitude of the time-averaged value of the Poynting vector is called the intensity I of the wave. 8

9 S = 1 μ E B I = S av = E maxb max 2μ = E 2 max 2μ c = 1 2 ε 2 E μ max = 1 2 ε 2 ce max Ch 32 review: 1. The nature of light: Light is an electromagnetic wave. A wave front is a surface of constant phase. A ray is a line along the direction of propagation, perpendicular to the wave fronts. The speed of light in vacuum is c. Light always travels more slowly in a medium with index of refraction n. n = c ; λ = λ v n 2. Reflection and refraction: At a smooth interface between two optical materials, the incident, reflected, and refracted rays and the normal to the interface all lie in a single plane called the plane of incidence. The angles of incidence, reflection, and refraction satisfy the law of reflection and the law of refraction. θ r = θ a n a sin θ a = n b sin θ b 3. Total internal reflection: When a ray travels in a material of greater index of refraction n a toward a material of smaller index n b, total internal reflection occurs at the interface when the angle of incidence exceeds a critical angle θ crit sinθ crit = n b n a 4. Polarization of light: The direction of polarization of a linearly polarized electromagnetic wave is the direction of the electric field. A polarizer passes waves that are linearly polarized along its polarizing axis and blocks wave polarized perpendicular to that axis. When polarized light of intensity I max is incident on a polarizer, the transmitted intensity is I = I max cos 2 φ Ch 34 review: 1. Reflection and refraction at a plane surface: When rags diverge from an object point P and are reflected or refracted, the directions of the outgoing rays are the same as though they had diverged from a point P called the image point. It they actually converge at P and diverge again beyond it, P is a real image of P; if they only appear to have diverged from P, it is a virtual image. Images can be either erect or inverted. 9

10 8. Lateral magnification: The lateral magnification m in any reflecting or refracting situation is defined as the ratio of the image height y to object height y. When m is positive, the image is erect; when m is negative, the image is inverted. 9. Focal point and focal length: The focal point of a mirror is the point where parallel rays converge after reflection from a concave mirror, or the point from which they appear to diverge after reflection from a convex mirror. The distance from the focal point to the vertex is called the focal length, denoted as f. 1. Object and image relationship: The object distance s and image distance s are related for a given imaging system. Plane mirror: 1 s + 1 s = (i.e. s = s ); m = s s = 1 Spherical mirror: = 2 = 1 s ; m = s s R f s Spherical refracting surface: n a + n b = n b n a ; m = n as s s R n b s 11. Sign rules: The following sign rules are used with all plane and spherical reflecting and refracting surfaces. s > when the object is on the incoming side of the surface (a real object); s < otherwise. s > when the image is on the outgoing side of the surface ( a real image); s < otherwise. R > when the center of curvature is on the outgoing side of the surface; R < otherwise. m > when the image is erect; m < when inverted. 12. Graphical methods for mirrors: Use the following principal rays A ray parallel to the axis A ray through (or proceeding toward) the focal point A ray along the radius 1

11 A ray to the vertex 13. Thin lenses: A thin lens with a positive focal length is called a converging lens or positive lens. A thin lens with a negative focal length is called a diverging lens or negative lens. The object-image relationship is the same for a converging lens as for a diverging lens, providing that the correct signs are used. 1 s + 1 s = 1 f m = s s (object-image relationship) (lateral magnification) 14. Graphical methods for lenses: Use the following principle rays A ray parallel to the axis A ray through the center of the lens A ray through (or proceeding toward) the first focal point Ch 35 review: 1. Interference: The overlap of waves from two coherent sources of monochromatic light forms an interference pattern. The interference pattern is determined by the principle of superposition, which states that the total wave displacement at any point is the sum of the displacements from the separate waves. 2. Two-source interference of light: When two sources are in phase, constructive interference occurs where the difference in path length from the two sources is zero or an integer number of wavelengths, i.e. d sin θ = mλ, (m =, ±1, ±2, ); destructive interference occurs where the path difference is a half-integer number of wavelengths, i.e. d sin θ = (m )λ, (m =, ±1, ±2, ). For small angles, the position of the mth bright fringe on the a screen is given by y m = R mλ d 11

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