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Final Examination PHY 250 Fall 2005 Instructions: 1. Express your answers in full, grammatically correct sentences using unambiguous, technical language. Avoid pronouns. 2. Do not place unexplained symbols or acronyms in the text. 3. The acceptable answers must be complete and, if applicable, with a convincing justification.. 4. Do not include irrelevant information. 5. Represent physical quantities by conventional symbols. 6. Use proper notation. Explain all symbols in the formulas unless asked otherwise. 7. Indicate vectors with arrows. (I will interpret the symbol without the arrow as the magnitude of the vector.) Grading policy: There are twenty topics on the final. Each topic contains four questions with progressive levels of difficulty (D, C, B, A). The highest-level question that is answered correctly determines the grade for the topic. For 85% of the topics (17) you must answer at least one question properly. For each grade you need to answer 60% of the questions (12) on the appropriate level. Topic 1 ELECTRIC CHARGE D. The law of charge conservation states that the electric charge of is always conserved. C. If the "charged" particles in a system (substance) cannot move we say that the system is B. Conductors are substances such that In metals, in electrolytes, the allow for transport of charge; fulfill this function. A. In general, charge density relates the charge of an object with its dimensions. Knowing the distribution of charge density, the charge Q of the object can be found directly from the definition. For a linear object with linear charge density λ(x), its charge is Q= Topic 2 ELECTRIC FIELD D. A line along which the electric field vector E is tangent to the line at each point, is called the . C. According to Coulomb’s law, the electrostatic force F21, that particle 1 exerts on particle 2 depends on the charges, q1 and q2, of the particles, and the relative position of the particles. + 2 + F21 = 1 (Mark schematically the force and the unit vector on the included figure.) By definition, the electric field vector E(r) at location r is a vector such that B. , + E - E is F= (Indicate the force at each particle.) A. Write an expression for the differential contribution to the electric field vector at point P due to the indicated fragment of the thin rod. Express the answer in terms of the variables marked in the figure and the linear charge density of the rod. y dE = P y x Mark this field on the figure. l L Topic 3 GAUSS’S LAW D. According to Gauss’s law, the net electric flux through any closed (Gaussian) surface is proportional to the net charge the surface. rr C. By definition, the flux of vector f (r ) over the indicated differential surface is rr r dΦ = f (r ) ⋅ dA r where dA is a vector v f dA the (differential) surface and its magnitude is equal to dA R Q B. Using the definition of electric flux, relate the flux over the indicated differential surface of the sphere to the magnitude E of the electric field (mark on the figure) produced by the charged particle located at the center of the sphere, the area of the surface, and the appropriate angle between these vectors. dΦE = A. Using Gauss’s law, relate the flux over the surface of the cylinder of radius R and height h, to the linear charge density λ of the uniformly charged rod. ΦE = R h Topic 4 ELECTRIC POTENTIAL such that the D. Electric potential V(r) at position r is a electric potential energy U(r) of a particle with charge q placed at this location would be U(r) = C. If electrostatic work performed on a charged particle moving from point A to point B along path 1 is W1 = 3.2 ⋅10-19 J, the work performed along path 2, which is twice as long, is 2 VA = 1V 1 VB = 3V W2 = B. The charge of the particle discussed in the previous question is Q= A. Write an integral expression for the electric potential at point P due to the charged thin rod. Express the answer in terms of the variables marked in the figure and the linear charge density of the rod. V= L P l x Topic 5 CAPACITORS D. Capacitance relates the charge transferred between the plates of the capacitor with the voltage C. The capacitance C of a capacitor depends on the geometrical shape of the capacitor and the properties of the dielectric between the plates C = κε0 A d where A is the area of the plates, d is the distance between the plates, and κ is the of the . B. The electric field between the plates of a 3V capacitor is approximately uniform. The broken lines in the figure (draw them) show the 1V and 2V equipotential surfaces and the solid lines are the electric field lines in this kind of a capacitor. 0V A. Since, in general, the electric field (vector) is opposite to the , of the electric potential, the voltage V between the plates of the capacitor separated by distance d yields a uniform electric field of strength A V E = . κ d Topic 6 ELECTRIC CURRENT D. The phenomenon of charge transport is referred to as the electric . C. The electric current through a , dq is defined by I = , where dq is the charge dt + + + + + + + + in time dt. r 1 r The average velocity v d ≡ ∑ v i of charge carriers, over a differential N i r vicinity of a given location r , is called the B. dA vd of the carriers at this location. θ vddt n A. The charge dq transferred in time dt, through the differential surface dA indicated in the figure, is dq = Hence, the current through the entire surface is equal to the over that surface I= r r J ∫ ⋅ dA . surface Topic 7 ELECTRICAL ELEMENTS D. A resistor is an electrical element with two sides for which (at any instant) the passing through this element (any cross section) is proportional to the potential difference (voltage) between its terminals. The proportionality coefficient R is called the of the resistor. C. A capacitor is an electrical element with two sides, called plates, for which the potential difference (voltage) between the plates is proportional to the charge Q Q Va -Q The proportionality coefficient C is called the Vb of the capacitor. Consistently with the figure Va - Vb = B. Voltage produced by a real source of electromotive force depends on its ε r Va ε and Vb r. Consistently with the figure I Va - Vb = A. A transistor is a semiconductor device with three terminals: , for which the resistance. current affects the Topic 8 ELECTRICAL CIRCUITS D. According to Kirchhoff’s junction rule, the sum of is equal to zero. According to Kirchhoff’s loop rule, is equal to zero. C. Consider a circuit containing an ideal seat of electromotive force and three resistors. Using Kirchhoff’s rules, write the necessary equations to find the currents indicated in the following figure I1 I3 R1 ε I B. R ε L I2 R2 R3 Consider a circuit containing an ideal seat of electromotive force, a resistor and an inductor. Using Kirchhoff’s loop rule, write the necessary differential equations to find the current in the loop A. For the same loop, write the equations for the complex current at frequency ω Topic 9 ALTERNATING CURRENT D. For a sinusoidal alternating current, it value I(t) is a sinusoidal function of time I (t) = Im sin (ωt + δI) where Im is called the of the current, ω is called the of the current, and δI is called the of the current. C. By definition, impedance Z of an element relates the across the element with the through the element by B. The number ϕω, relating the with is called the phase angle between the current and the voltage. A. Complex impedance Zω relates the complex voltage with the by V(t) = If the complex impedance of a system has (complex) value Zω, the impedance of the system is Zω = and the phase angle between the voltage across the system and the current through the system is ϕω = Topic 10 RLC CIRCUIT D. The average electrical power delivered to an RLC circuit depends on the through the circuit, the across the circuit, and the Pav = Im Vm cosϕ . C. In the given RLC circuit, the voltage across the resistor is in phase with the current in the circuit while the voltage across the inductor current by voltage across the capacitor by 90°. 90° and the R L C current B. In a given series RLC circuit the resonance occurs at the frequency for which the I V assumes its lowest value. Therefore ω0 = A. If at resonance the rms value of the voltage across each element is 1V, the rms value of the voltage across the AC source is Vrms = Topic 11 ELECTRICAL CIRCUITS IN D. Identify the circuit and explain its purpose. R2 OUT R1 C. Identify the circuit and explain its purpose. IN OUT R C B. Identify the circuit and explain its purpose. + OUT IN +5 A. Identify the circuit and explain its purpose. A B X Topic 12 MAGNETIC INTERACTION D. According to the definition of the magnetic field vector, the magnetic force FB (mark on the figure), exerted on a and particle, depends on the B v + of the particle, and the by FB = B C. Since the magnetic work (performed on the particle) is zero. B. The magnetic potential energy of an object with magnetic moment μ (mark on the figure) placed in the magnetic field B is N UB = B B S According to this expression the potential energy degrees. has a zero value when the angle between the two vectors is A. Since the differential magnetic force (mark on the figure) exerted on a differential fragment of the wire has the value dFB = , B the magnetic force exerted on a circular loop of radius R is I B FB = B . Topic 13 MAGNETIC FIELD VECTOR D. According to Gauss’s law, the net magnetic flux ΦB over B has a value equal to . M z N C. On the included figure, mark the direction of the magnetic field at points K, M and N. I y K x B. The Biot-Savart law allows one to determine the magnetic field produced by arbitrarily shaped conducting wire carrying electrical current I. According to this law, the contribution dB to the magnetic field from a differential fragment “ds” of the wire is dB = where ds is a A. In the included figure, a uniform magnetic field of known value B, is in the plane of the rectangle with dimensions given in the figure. Determine the linear integral of the field along this rectangle (closed loop). B b a Topic 14 SOURCES OF MAGNETIC FIELD D. Mark the magnetic field lines for the field produced by the bar magnet. N C. I Mark the magnetic field lines for the field produced by the solenoid. Mark the location of the magnetic poles of the solenoid. I B. The magnetic field outside the solenoid is approximately zero (vector). Inside the solenoid the field is uniform and its strength B0 = B where I is the current . and the n A. Permanent magnets are made from ferromagnetic substances because they exhibit the of magnetization. As seen in the included figure (complete the drawing), the magnetization (magnetic field of the substance) is sustained even after the . Topic 15 MAGNETIC INDUCTION D. Explain the difference between magnetic induction, inductor and inductance. Inductor is , induction is , . and inductance is C. According to Faraday’s law of induction ε= −N the dΦ B dt induced in a coil consisting of N loops is proportional to the rate of change in the over the surface of the loop. v B. In the figure, the magnet is moving away from the ring. Mark the direction of the (positive) magnetic flux over the surface of the ring and the direction of the (positive) current induced in the ring. N A. When a coil, consisting of N loops, rotates in a magnetic field (as in an electrical generator), the magnetic flux over the surface of this coil is a sinusoidal function of time Φ(t) = ΦB,maxcos(ωt). Use Faraday’s law of induction to derive an explicit function for the induced electromotive force. ε(t) = . Topic 16 ELECTROMAGNETIC RADIATION D. For a certain electromagnetic radiation (a superposition of monochromatic waves), the function relating the intensity of each sinusoidal wave with the wavelength is called of that radiation. C. Dispersion is an effect in which light is . B. Formally defined as the average magnitude of the Poynting vector, the intensity of the radiation dU I = Sav = dA ⋅ dt represents the amount of energy passing of area dA in time dt. A. The plane wave solutions to the wave equation have the form E (x , t ) = E m cos(kx − ωt ) . This wave has a frequency y f= and wavelength x λ= Mark the direction of wave propagation. z Topic 17 REFLECTION AND REFRACTION D. Identify the rays marked in the figure. Mark the angle of incidence, the angle of reflection and the angle of refraction. C. Snell’s law of refraction states that the angle of refraction and the angle of incidence are related by n1 n2 where n1, and n2 are the refraction indexes of the two media at the refracting interface. B. The critical angle θc is an angle of incidence at which . The critical angle depends on the indexes of refraction of the media at the interface sin θc = . A. The index of refraction n of a medium is defined by n≡ where . Topic 18 MIRRORS D. The location s of the object and the location s’ of the image both with respect to the mirror are related by the mirror equation where f is the C. Draw a ray diagram to find the image I produced by a plane mirror from a real object O. Is the image real or virtual? The image is O . B. Draw a ray diagram to find the image I produced by a concave mirror from a real object O. Is the image real or virtual? O The image is F A. Draw a ray diagram to find the image I produced by a diverging lens from a virtual object O. Is the image real or virtual? The image is F F O Topic 19 LENSES D. For a thin lens, the position of an object (s) and the position of the image (s’) is related by the thin lens equation: where f is the C. The magnification of an optical element or instrument is defined as the A negative value of magnification means that . Mark these dimensions on the figure. B. Identify all rays marked on the figure. O Fi Fo I A. Abberation of an element is a . Spherical aberration results from of the lens. Chromatic aberration results from of the lens. Topic 20 THE LAB EXPERIMENTS D. An ammeter should be connected in in which the current is measured. with the element with the element A voltmeter should be connected in across which the current is measured. C. When a Weatston Bridge is balanced, the current in the galvanometer has a value. The resistance of the investigated resistor is ralated to the known resistances R1, R2, and R3 by R2 X G R3 R4 X= B. Hall effect is used to measure magnetic fields. An electric field, transverse to is produced. A. According to the law of Malus, the intensity of the light transmitted through a polarizer, depends on the intensity of the polarized light incident on the polarizer I = I0 cos2 θ where θ is the angle between .