r 0 at infinity p gy v p gy v v arad r Newton s second law dt Solution for above equation: T For a spring mass oscillator, m For a simple pendulum,

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1 Scalar Product: B B B B B Vector Product: B B sin cos B x x y y z z B B B iˆ y z z y B B ˆj z x x z B B kˆ x y y x Equations of motion: v v at x x vt at v v a( x x) Radial cceleration: v arad r Newton s second law F ma Magnitude of kinetic friction F Ff k k N Magnitude of static friction F F f s N s Definition of work W F dx Definition of kinetic energy: KE mv Change in gravitational potential energy: U mg y g Elastic potential energy: Uel kx Work-Energy Theorem: W U KE Center-of-mass position n X COM ximi M i Definition of momentum p mv Conservation of momentum p i p f Definition of torque r F Newton s second law for rotation I Conditions for rolling: acom R and vcom R ngular momentum: L r p or L I, where I miri Newton s Law of Gravitation: Gmm Gmm FG and U G with r r UG at infinity Bernoulli's Equation: p gy v p gy v Equation for Simple Harmonic Motion: d x x dt Solution for above equation: x t cos t Where, f T For a spring mass oscillator, k m For a simple pendulum, g L Wave Equation: y x, t y x, t x v t Solution to above equation: yx, t cos kx t Where, k, f, v f Standing waves on fixed string: y x, t sin kx sin t v fn n L SW i

2 Doppler Effect: v vl fl fs v vs o 5 o T C T F 3 9 T K T o C 73.5 L L T V VT Q mct nct Q F V ml F V dq H k TH TC dt L pv nrt 3 Ktr nrt 3 CV R ideal monatomic gas 5 CV R ideal diatomic gas w/o vibration V W pdv V U Q W U ncv T for ideal gas pv const adiabatic process TV const W Q e Q Q e Carnot H C T T S C dq T H H S k ln w R 8.34 J mol K 3 N 6. molecules mole atm = 35 N / (m ) =.x 5 Pa /4πε = 8.99 x 9 Nm /C e = -.6 x -9 C qq F ˆ E qe r 4 r qenc E E d V V E dl b a U q V b a ˆ V ˆ E i j V kˆ V x y z Q CV series C C C C eq 3 Ceq C C C3 parallel U CV ue E E E K dq I dt J nqv d E J L R V IR P VI Req R R R3 series parallel R R R R eq 3 q C e t RC charging q Qe t RC discharging F qv B B Bd df Idl B B, NI

3 U B ˆ qv r B 4 r ˆ Idl r db 4 r Bdl I enc B dl i i C D db dl dt de id dt di di M and M dt dt NB NB M i i di L, dt V s N s Vp N p N B L i U LI ue B, di Rt e L dt L LC IRMS I for i I cos t VRMS V for v V cos t V IR R V IX, where X L L L L V IX, where X C C C C V IZ, where Z R X L X C P vg VI cos, X tan L X R C

5 Physics 6- Spring 4 Exam 3 Name: Box# Multiple Choice (5 points each): ) tube of mercury with resistivity Ωm has an electric field inside the column of mercury of magnitude 8 V/m that is directed along the length of the tube. How much current is flowing through this tube if its radius is 6. mm? ) 4.8 B) 6. C). D).3 E) 9. F) 5. G) 55.4 H) 87.3 I) 5 J) 34 E 8V m J m m and I J. m 6 m 5. ) The emf and the internal resistance of a battery are as shown in the figure. When the terminal voltage Vab is equal to. V, what is the current through the battery? ). B) 4.3 C) 5. D). E) 5.5 F) 9. G) 3. H) 6. I) 5. J) 4 V.V E Ir 3V I ab I 5.

6 3) proton moving in the positive x direction enters a magnetic field. If the proton experiences a magnetic deflection in the negative y direction, the magnetic field in this region is ) in the direction of the +x axis. B) in the direction of the -x axis. C) in the direction of the +y axis. D) in the direction of the -y axis. E) in the direction of the +z axis. F) in the direction of the -z axis. G) in any direction perpendicular to the proton velocity. H) zero. I) undefined. FB qv B. Since the proton is positively charged, the direction is in the same direction given by the right hand rule. 4) If the current density in a wire of radius R is given by J = J + kr, < r < R, what is the total current in the wire? ) kr / B) kr C) J R + kr / D) J R + kr 3 /3 E) J πr + kπr 3 /3 F) J πr + kπr / G) J πr + kr / H) kπr 3 /3 I) kr 3 /3 J) J (kπr 3 /3) I J r d J r rdr J kr rdr J rdr kr rdr R R J rdr kr dr 3 J R k R 3 R R R R R

7 5) The figure shows the cross-section of a hollow cylinder of inner radius a =. cm and outer radius b =. cm. uniform current density of. / cm flows through the cylinder parallel to its axis. Calculate the magnitude of the magnetic field at a distance of d =. m from the axis of the cylinder. (μ = 4π -7 T m/) ). x -5 T B).5 x -5 T C). x -6 T D).9 x -6 T E) 6.3 x -6 T F) 9.7 x -6 T G). x -7 T H) 4.8 x -7 T I) 9. x -7 T J) T In this case, J is uniform, so, I J J b a cm b a cm 4cm cm 9.4 Then, from symmetry, we use mpere s law to find that: Bd s i enc 7 T m Br B.9 6 T 6) Calculate the current through a.-m long gauge (having radius.3 mm) nichrome wire if it is connected to a 3.-V battery. The resistivity of nichrome is -8 Ω m. ) B) C) 3 D) 4 E) 5 F) 6 G) 7 H) 8 I) 9 J) 8 L m m R m and, V IR V 3V I R 3.

8 7) The figure shows three identical light bulbs connected to a battery having a constant voltage across its terminals. What happens to the brightness of light bulb 3 when the switch S is closed? ) Momentarily goes up then back to its original brightness. B) Momentarily goes down then back to its original brightness. C) Permanently gets brighter. D) Permanently gets dimmer. E) No change. 8) current is running through a wire next to the circuit shown in the figure with the switch S open and the capacitor uncharged. The battery has no appreciable internal resistance. Which one of the following graphs best describes the magnitude of the force on the wire as a function of time t after closing the switch? F F F F B C D F E

9 9) For the circuit shown in the figure, determine the current in the 4.-Ω resistor. ). B). C).4 D).6 E).8 F). G). H).4 I).6 J).8 We first find the equivalent resistance of the three resistors in parallel: 4 Req R 5, then the equivalent resistance of eq... 4 the three in series: Req. 4 5 and, then the current through the entire circuit: I = V/R = 3. Then, the voltage across the resistors in parallel is /5V and so the current through the 4.-Ω resistor is 3/5. ) Consider the circuit shown in the figure. Note that two currents are shown. Calculate the emf V ε3. IR V 3V I R 3... ) 5 V B) 48 V C) 44 V D) 4 V E) 4 V F) 38 V G) 36 V H) 34 V I) 3 V J) 3 V V.V E Ir 3V I ab I 5.

10 ) For the circuit shown in the figure, the switch S is initially open and the capacitor voltage is 8 V. The switch is then closed at time t =. What is the charge on the capacitor when the current in the circuit is 3 μ? ) μc B) 4 μc C) μc D) μc E) 8 μc F) 33 μc G) 39 μc H) 43 μc I) 47 μc J) 5 μc t RC dq Q q Qe t RC and i e. dt RC Now, at some time, i = 3μ, or: Q t RC i e 3 RC t RC RC RC R e Q CV V then, t RC R q Qe Q 3 C R 3 43C V ) For the circuit shown in the figure, the capacitors are all initially uncharged, the connecting leads have no resistance, the battery has no appreciable internal resistance, and the switch S is originally open. Just after closing the switch S, what is the current in the 5.-Ω resistor? 3. ) B). C).4 D).6 E).8 F). G). H).4 I).6 J).8 Immediately after the switch is closed, the voltage drop across the capacitor is V, so no current through the 5 Ω resistor.

11 3) fter the switch S has been closed for a very long time, what is the potential difference across the 8.-μF capacitor? 3. ). V B) 4.3 V C) 5. V D). V E) 5. V F) 9. V G) 3. V H) 6. V I) 3. V J) 4 V For long times, no current passes through the capacitors, so the current just goes through the three resistors in series. The equivalent resistance is just 3 Ω, with 3V means is passing through each resistor. Then the voltage drop across the 5 Ω resistor is just 5 V. 4) charge is accelerated from rest through a potential difference V and then enters a uniform magnetic field oriented perpendicular to its path. The field deflects the particle into a circular arc of radius R. If the accelerating potential is doubled to V, what will be the radius of the circular arc? ) R B) R/ C) 4R D) R/4 E) R F) R / G) R H) 3R I) 3 R J) R/ 3 KE mv qv and qv v m F qv B B v qvb ma m R mv mv m qv mv R qvb qb qb m B q

12 5) circular coil of wire of turns and diameter. cm carries a current of.. It is placed in a magnetic field of T with the plane of the coil making an angle of 45 with the magnetic field. What is the magnetic torque on the coil? ). Nm B). Nm C).4 Nm D).6 Nm E).8 Nm F) 3. Nm G) 3. Nm H) 3.4 Nm I) 3.6 Nm J) 3.8 Nm B and NI..m 3.4 m so B sin 3.4 m T sin 45.Nm 6) negatively charged particle is moving to the right, directly above a wire having a current flowing to the right, as shown in the figure. In which direction is the magnetic force exerted on the particle? ) up B) down C) out of the page D) into the page E) no magnetic force The B-field from the wire is coming out of the page at the charge, and from F qv B, with q negative, then the force is upwards.

13 7) large number of very long wires of diameter mm are laid side-by-side to form a plane. If.5 of current is passed through each wire (in the same direction), what is the magnitude of the magnetic field cm above (and in the middle of) the plane? (μ = 4π -7 T m/) ).6 x -4 T B) 5. x -4 T C) 7.9 x -4 T D).6 x -3 T E) 4.7 x -3 T F) 6.6 x -3 T G) 9. x -3 T H). x - T I) 4.4 x - T J) 6.5 x - T Bd s i enc BL 4 B 7 L T T m T m m m

Induced voltages and Inductance Faraday s Law

Induced voltages and Inductance Faraday s Law concept #1, 4, 5, 8, 13 Problem # 1, 3, 4, 5, 6, 9, 10, 13, 15, 24, 23, 25, 31, 32a, 34, 37, 41, 43, 51, 61 Last chapter we saw that a current produces a magnetic