Physics 1202: Today s Agenda Announcements: Lectures posted on: www.phys.uconn.edu/~rcote/ Office hours: Monday 2:30-3:30 Thursday 3:00-4:00 Homework #2: due this coming Friday/ Labs: Already begun last week Policy on clicker questions 80 % of total points gives 100% No make-up for missed clicker questions Policy on Homework Lowest homework will be dropped No extension Today s Topic : Chapter 21: Electric current & DC-circuits eview» Electric current, resistance, Ohm s law & power» esistance in series & parallel» Kirchhoff s rules» Capacitances in series & parallel C-circuits Measuring devices Chapter 22: Magnetism Magnetic field (B) & force Motion of a charged particle in B-field 1
e e = 21-1: Electric current = DQ / Dt 21-2: esistance & Ohm s Law esistance esistance is defined to be the ratio of the applied voltage to the current passing through. What does it mean? it is the a measure of the friction slowing the motion of charges Analogy with fluids UNT: OHM = W V 2
21-3: Energy & Power Batteries & esistors Energy expended chemical to electrical to heat ate is: What s happening? Charges per time Assert: Energy drop per charge For esistors: Units: e e = 3
esistors in series the current is the same in both 1 and 2 the voltage drops add Summary V 1 esistors in parallel the voltage drop is the same in both 1 and 2 the currents add 2 V 1 2 e 1 1 e 2 2 3 e 3 4
a V Capacitors in Parallel Q 1 Q 2 º V a Q b b Capacitors in Series +Q -Q a b º Þ C = C 1 + C 2 +Q -Q a b Þ 21-7: C Circuits Consider the circuit shown: a What will happen when we close the switch? Add the voltage drops going around the circuit, starting at point a. V C b c + Q/C V = 0 n this case neither nor Q are known or constant. But they are related, This is a simple, linear differential equation. 5
Case 1: Charging C Circuits a Q 1 = 0, Q 2 = Q and t 1 = 0, t 2 = t V C b c To get Current, = dq/dt Q t t C Circuits Case 2: Discharging: Q 1 = Q 0, Q 2 = Q and t 1 = 0, t 2 = t To discharge the capacitor we have to take the battery out of the circuit (V=0) V a C b c To get Current, = dq/dt Q t t 6
, ACT 1 Consider the simple circuit shown here. nitially the switch is open and the capacitor is charged to a potential V O. mmediately after the switch is closed, what is the current? V a C b c A) = V O / B) = 0 C) = C D) = V O / exp(-1/c) 21-8: Electrical nstruments The Ammeter The device that measures current is called an ammeter. 1 2 - + A e deally, an ammeter should have zero resistance so that the measured current is not altered. 7
Electrical nstruments The Voltmeter The device that measures potential difference is called a voltmeter. 2 1 2 v e V An ideal voltmeter should have infinite resistance so that no current passes through it. Problem Solution Method: Five Steps: 1) Focus on the Problem - draw a picture what are we asking for? 2) Describe the physics - what physics ideas are applicable - what are the relevant variables known and unknown 3) Plan the solution - what are the relevant physics equations 4) Execute the plan - solve in terms of variables - solve in terms of numbers 5) Evaluate the answer - are the dimensions and units correct? - do the numbers make sense? 8
Example: Power in esistive Electric Circuits A circuit consists of a 12 V battery with internal resistance of 2 W connected to a resistance of 10 W. The current in the resistor is, and the voltage across it is V. The voltmeter and the ammeter can be considered ideal; that is, their resistances are infinity and zero, respectively. What is the current and voltage V measured by those two instruments? What is the power dissipated by the battery? By the resistance? What is the total power dissipated in the circuit? Comment on these various powers. Step 1: Focus on the problem Drawing with relevant parameters Voltmeter can be put a two places V What is the question? What is? What is V? What is P battery? What is P? What is P total? Comment on the various P s V r e 2 W 12 V 10 W A 9
Step 2: describe the physics What concepts are relevant? Potential difference in a loop is zero Energy is dissipated by resistance What are the known and unknown quantities? Known: = 10 W,r = 2 W, e = 12 V Unknown:, V, P s Step 3: plan the solution What are the relevant physics equations? Kirchoff s first law: Power dissipated: For a resistance 10
Step 4: solve with symbols Find : e - r - = 0 Find V: r e Find the P s: A Step 4: solve numerically Putting in the numbers 11
Step 5: Evaluate the answers Are units OK? [ ] = Amperes [ V ] = Volts [ P ] = Watts Do they make sense? the values are not too big, not too small total power is larger than power dissipated in» Normal: battery is not ideal: it dissipates energy 12
Magnetism Magnetic effects from natural magnets have been known for a long time. ecorded observations from the Greeks more than 2500 years ago. The word magnetism comes from the Greek word for a certain type of stone (lodestone) containing iron oxide found in Magnesia, a district in northern Greece or maybe it comes from a shepherd named Magnes who got the stuff stuck to the nails in his shoes Properties of lodestones: could exert forces on similar stones and could impart this property (magnetize) to a piece of iron it touched. Small sliver of lodestone suspended with a string will always align itself in a north-south direction. ie can detect the magnetic field produced by the earth itself. This is a compass. Bar Magnet Bar magnet... two poles: N and S Like poles repel; Unlike poles attract. Magnetic Field lines: (defined in same way as electric field lines, direction and density) You can see this field by bringing a magnet near a sheet covered with iron filings Does this remind you of a similar case in electrostatics? 13
Electric Field Lines of an Electric Dipole Magnetic Field Lines of a bar magnet 14
Magnetic Monopoles One explanation: there exists magnetic charge, just like electric charge. An entity which carried this magnetic charge would be called a magnetic monopole (having + or - magnetic charge). How can you isolate this magnetic charge? Try cutting a bar magnet in half: S N S N S N n fact no attempt yet has been successful in finding magnetic monopoles in nature. Many searches have been made The existence of a magnetic monopole could give an explanation (within framework of QM) for the quantization of electric charge (argument of P.A.M.Dirac) Source of Magnetic Fields? What is the source of magnetic fields, if not magnetic charge? Answer: electric charge in motion! eg current in wire surrounding cylinder (solenoid) produces very similar field to that of bar magnet. Therefore, understanding source of field generated by bar magnet lies in understanding currents at atomic level within bulk matter. Orbits of electrons about nuclei ntrinsic spin of electrons (more important effect) 15
22-2:Forces due to Magnetic Fields Electrically charged particles come under various sorts of forces. As we have already seen, an electric field provides a force to a charged particle, F = qe. Magnets exert forces on other magnets. Also, a magnetic field provides a force to a charged particle, but this force is in a direction perpendicular to the direction of the magnetic field. Definition of Magnetic Field Magnetic field B is defined operationally by the magnetic force on a test charge. (We did this to talk about the electric field too) What is "magnetic force"? How is it distinguished from "electric" force? Start with some observations: Empirical facts: a) magnitude: µ to velocity of q q F mag v b) direction: ^ to direction of q 16
Magnetic Force on a Moving Charge When moving through a magnetic field, a charged particle experiences a magnetic force where B is called Magnetic Field: t is a vector quantity o The S unit of magnetic field is the Tesla (T) The cgs unit is a Gauss (G) o 1 T = 10 4 G Earth B-field: 0.5 G or 5 x 10-5 T Direction of Magnetic Force Given by the right-hand rule direction of F mag on a positive charge F mag on a negative charge:opposite direction 2017 Pearson Education, nc. 17
Direction of Magnetic Force This relationship between the three vectors magnetic field, velocity, and force can also be written as a vector cross product: max if v & B perpendicular 0 if v & B parallel ight Hand ule: Your thumb points in the direction of the force, F, for a positive charge 22-3: motion of a charges particle Consider a positive charge in an electric field force in the direction of field E in a magnetic field force is perpendicular to field B This leads to very different motions Because F mag is perpendicular to the direction of motion, the path of a particle is circular Also, while E can do work on a particle, B cannot the particle s speed remains constant 2017 Pearson Education, nc. 18
Lorentz Force The force F on a charge q moving with velocity v through a region of space with electric field E and magnetic field B is given by: B v F q Units: B v q F 1 T (tesla) = 1 N / Am 1G (gauss) = 10-4 T B v q F = 0 1A, ACT 2 Two protons each move at speed v (as shown in the diagram) toward a region of space which contains a constant B field in the -y-direction. What is the relation between the magnitudes of the forces on the two protons? (a) F 1 < F 2 (b) F 1 = F 2 (c) F 1 > F 2 y z 1 2 v v x B 1B What is F 2x, the x-component of the force on the second proton? (a) F 2x < 0 (b) F 2x = 0 (c) F 2x > 0 19
Circular motion Force is perp. to v q = 90 o so sinq = 1 or F=qvB Work proportional to cos f (recall 1201)! f :angle between F and Dx cos f =0 (perpendicular) W=0 Þ DK=0 Kinetic energy not changed Velocity constant: UCM!, ACT 3 Cosmic rays (atomic nuclei stripped bare of their electrons) would continuously bombard Earth s surface if most of them were not deflected by Earth s magnetic field. Given that Earth is, to an excellent approximation, a magnetic dipole, the intensity of cosmic rays bombarding its surface is greatest at the (The rays approach the earth radially from all directions). A) Poles B) Equator C) Mid-lattitudes 20
Motion of Charged Particles f a v makes an angle with B the component of v along B will not change a particle with initial v at an angle to B will move in a helical path. Trajectory in Constant B Field Suppose charge q enters B field with velocity v as shown below. (v^b) What will be the path q follows? v F x x x x x vx F q Force is always ^ to velocity and B. What is path? Path will be circle. F will be the centripetal force needed to keep the charge in its circular orbit. Calculate : B 21
adius of Circular Orbit Lorentz force: centripetal acc: Newton's 2nd Law: v F x x x x x vx F q B Þ Þ This is an important result, with useful experimental consequences! atio of charge to mass for an electron e - 1) Turn on electron gun 2) Turn on magnetic field B DV gun 3) Calculate B next week; for now consider it a measurement 4) earrange in terms of measured values, V, and B Þ & 22
Lawrence's nsight " cancels " We just derived the radius of curvature of the trajectory of a charged particle in a constant magnetic field. E.O. Lawrence realized in 1929 an important feature of this equation which became the basis for his invention of the cyclotron. Þ ewrite in terms of angular velocity w! Þ Þ does indeed cancel in above eqn. So What?? The angular velocity is independent of!! Therefore the time for one revolution is independent of the particle's energy! We can write for the period, T=2p/w or T = 2pm/qB This is the basis for building a cyclotron. The Hall Effect l v d F c B Force balance d qe H - c Hall voltage generated across the conductor B a Using the relation between drift velocity and current we can write: 23
22-4: Magnetic Force on a Current Consider a current-carrying wire in the presence of a magnetic field B. There will be a force on each of the charges moving in the wire. What will be the total force DF on a length Dl of the wire? Suppose current is made up of n charges/volume each carrying charge q and moving with velocity v through a wire of crosssection A. Force on each charge = Total force = N S Current = Þ Simpler: For a straight length of wire L carrying a current, the force on it is: or 24
, ACT 4 A current flows in a wire which is formed in the shape of an isosceles triangle as shown. A constant magnetic field exists in the -z direction. What is F y, net force on the wire in the y- direction? y (a) F y < 0 (b) F y = 0 (c) F y > 0 x ecap of Today s Topic : Chapter 21: Electric current & DC-circuits eview» Electric current, resistance, Ohm s law & power» esistance in series & parallel» Kirchhoff s rules» Capacitances in series & parallel C-circuits Measuring devices Chapter 22: Magnetism Magnetic field (B) & force Motion of a charged particle in B-field 25