Physics 112 Formulae
area of circular ring of radius r and thickness dr=2 π r dr
volume of "meat" of cylindrical shell of length L, radius r, and thickness dr=2 π r L dr
volume of "meat" of spherical shell of radius r and thickness dr=4 π r2 dr
UNIT 2: Potential Energy, Voltage, Capacitors
∆PE=qo∆V or PE=qoV
PE=U=kq0q1r01=14πεo q0q1r01, V=kq0r0=14πεo q0r0 (V→0 as r→∞);
PE=U=q04πεo q1r1+q2r2+q3r3+…, V=14πεo q1r1+q2r2+q3r3+… (V→0 as r→∞);
PE=q04πεoabdqr, V=14πεoifdqr V→0 as r→∞;
PEb-PEa=-abF∙dl=-abq0E∙dl; Vb-Va=-abE∙dl;
Infinite sheet: V=Ed
Q=CV; for parallel plates, C=Aεod;
PE=12CV2=Q22C=12QV;
Ceq=C1+C2+…; 1Ceq=1C1+1C2+…;
Cnew=KCold;
UNIT 3: Current, Resistance, Introduction to Circuits
I=dQdt=nqvdA; J=IA=nqvd; J=nqvd;
Ohm’s Law: E=ρJ and V=IR; R=ρLA;
P=IV=I2R=V2R;
Req=R1+R2+R3+…; 1Req=1R1+1R2+1R3+…;
Iin=Iout; closed loop∆V=0;
Charging: Qt=Qf1-e-tRC=CVo1-e-tRC; It=VoRe-tRC;
Discharging: Qt=Qoe-tRC; It=QoRCe-tRC;
UNIT 4: Magnetic Fields
F=qv×B (F=qvBsinθ); r=mvqB ; ΦB=B⋅dA=B dA cosθ;
F=IL×B F=ILB sinθ; ΦB=BAcosθ (in certain situations)
dF=I dL×B dF=I dL B sinθ; ΦB=B⋅dA=0 (closed surface)
μ=NIA; τ=μ×B (τ=μBsinθ)
B=μo4πqv×rr2; dB=μo4πI dl×rr2;
B⋅dl=μoIthru+Id where Id=εodΦEdt;
B=μoI2πr; B=μonI=μoNLI; B=μoI2r;
V=-dΦBdt; E=cB; c=1εoμo;
I=12εocEmax2=EmaxBmax2μo=Emax22μoc=12εoμoEmax2;
Physics 112 – Test #4
14 November 2016
Name______
Directions: This is a CLOSED BOOK test, and you may only use the calculator and cheat sheet provided. SHOW ALL OF YOUR REASONING! You can only get partial credit for an incorrect answer if you show your reasoning. You may have as much time as you like to finish this exam.
1. (5 pts) A bunch of small bar magnets and a long, straight wire are in deep space not moving. The magnitude of the magnetic moment of each bar magnet is as shown, as well as the current in the wire. Using Gauss’s Law for magnetism, determine the net magnetic flux out through the spherical closed surface (radius = 10 m) drawn as shown. (Trick question?)
2. (10 pts) Remember your professor’s speaker demo—the first one? He put a folded piece of paper with a wire taped to it in between the poles of a magnet. When electrical currents slosh back and forth in the wire at the same frequencies as the music being played, you could hear the music. Explain, using the physics you learned in this unit, why this happens. (You may refer to the set up demo at the side of the room.)
3. A long wire carrying a current of 12.0 A is bent into the shape of a right triangle of 1 “turn” as shown. It is placed into a constant uniform external magnetic field B = 20.0 T in such a way that the lines of B are parallel to the plane of the right triangle and are perpendicular to the right triangle’s hypotenuse.
a) (11 pts) Determine the magnitude and direction of the magnetic moment of the current-carrying right triangle wire.
b) (11 pts) Determine the magnitude and direction (or sense) of the torque this right triangle wire feels because it is in an external magnetic field.
4. (17 pts) A long, straight solenoid with a cross-sectional area of 0.0055 m2 is wound with 500 turns of wire per centimeter, and the windings carry a current of 0.50 A. A second winding of 5 turns encircles the solenoid at its center. The current in the long solenoid goes (linearly) from 0.50 A to 5.50 A in 0.010 seconds. What is the induced voltage in the second (outer) winding during this time? The outer winding has a cross-sectional area of 0.0075 m2.
5. (17 pts) A current winds around as shown at right. Starting with the Biot-Savart law (and NOT a formula on your cheat sheet), derive an expression for the magnetic field at point P. Evaluate all integrals (they are easy). Your expression may include I, a, b, and fundamental constants. Indicate the direction of the magnetic field at P.
6. (17 pts) A long straight wire of radius 0.00050 m carries a current density that varies with distance from the middle of the wire as Jr=Ar, where A is a constant equal to 1.6 × 105 A/m2. In the picture at right, the current comes out of the page. Use Ampère’s Law to determine the magnitude and direction of the magnetic field at point P in the diagram at right. Evaluate all integrals (they are easy), and the answer might surprise you.
7. (12 pts) The figure at right shows a circular region of radius 0.050 m where E points into the page. This electric field is INCREASING into the page at a rate of 106 N/C each second. What is the magnitude and direction of the induced magnetic field at point P, 0.10 m above the center of the circular region? Explain.
BONUS BRAIN BUSTER!!! (+5 points)
A thin aluminum (not magnetizable) bar 0.10 m long is pulled at a constant speed of 500 m/s through a constant magnetic field of 20 T (!) as shown. Determine the voltage difference between the top of the bar and the bottom of the bar. (HINT: You may assume that the electric field in the bar is uniform.)
Credit will only be given for correct reasoning. If you are able to guess the correct answer but give no correct reasoning, you will receive no extra credit. However, if you give some correct reasoning that would (or does) lead to the correct answer, at least some credit will be given.
______
By signing my name above, I affirm that this test represents my work only, without aid from outside sources. In all aspects of this course I perform with honor and integrity.