# 3/31/2014. Resistors in series. Resistors in parallel. Ohm s Law. Review for Test 2. Electric Power (cont d) V IR. R constant I

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1 Ohm s Law eview for Test Ohm s law states that the current flowing through a piece of material is proportional to the voltage applied across the material. The resistance () is defined as the ratio of to. constant Circuit diagram of the flashlight, and illustration of the variations in Ohm s Law: 1 The S unit of esistance is olts/ampere (/A) or Ohms (). Electric Power (cont d) The power derived above, P = is the power that must be delivered to an object in order to support the flow of current through that object while maintaining a voltage of across it. Below are alternative expressions for power, depending on whether the voltage or the current is known: P P ( ) 3 esistors in series 1 1 Equivalent circuit When resistors are in series they are arranged in a chain, the current has only one path to take. So, the current is the same through each resistor in series. At the same time, the sum of the potential drops across each resistor must equal to the total potential drop of the voltage supply across the whole chain, i.e., the voltage supply s emf,. Thus, = 1 + eq = 1 + eq = 1 +. We can generalize the above result to any number of resistors, so the equivalent resistance of resistors in series is: eq = and the loop rule is: = = 1 + e q 4 esistors in parallel When resistors are arranged in parallel, the current has multiple paths to take. Nevertheless, the currents of different paths must add to equal the total current entering the parallel combination, i.e., 1 + =. At the same time, each resistor is connected completely across the voltage supply. So, the potential difference across each resistor is the same equal to the emf of the voltage supply, 1 = =. 1 eq 1 Equivalent circuit 1 1 eq 1 = 1 + The next questions make use of the following equations: For more than two resistors connected in parallel, eq

2 0a. Adding an open circuit in parallel 0b. Adding an open switch in parallel S Question: What is the equivalent resistance of the above parallel connection if switch S is open? Question: What is the equivalent resistance of the circuit? Ans. eq = Ans. eq = / 7 8 0c. Comparing oltages 0d. Comparing Current f 1 = /, rank the voltages 1, and = = 3. 1 > > > = = > 1 Ans. Number 1. First, = 3 because they are voltages across resistors connected in parallel. Secondly, their equivalent resistance is /. Because 1 is also /, 1 and the parallel connection should split equally, i.e., the 9 voltages 1,, 3 should all be /. n the above, if we take away, the current through 1 will 1. increase. decrease 3. make no change 4. There is insufficient information to tell. Ans. Number. Taking away will increase the equivalent resistance of the parallel connection. This will cause the total resistance of the circuit to increase, so will cause the total current, which is the same current passing through 1, to 10 decrease. 0e. Comparing power Forces on a Moving Charge Particle due to a Magnetic Field The Lorentz or magnetic force is given by: Suppose that all the resistors 1,, to 5 are the same. Select the statement that is correct has the biggest power and the power through 4 is equal that of 5 and is the lowest.. The power through is the same as 3 and is higher than that of 5 and The power through is the same as that through 3 is the lowest of all. F = qvbsin where q is the charge of the particle, v is the particle s speed, B is the magnetic field and is the angle between the B and v vector. n S units, magnetic fields are in Tesla. As an example, the earth s magnetic field is about 10-4 Tesla in the vicinity of its surface. vsin +q 0 F v B B B 11 The direction of the Lorentz force is determined by a right-hand rule (H). n this class note, we refer to this right-hand rule as H-1. Later in this chapter, we will encounter another H! 1

3 ight Hand ule No. 1 ight Hand ule No. 1. (H-1) Extend the right hand so the fingers point along the direction of the magnetic field and the thumb points along the velocity of the charge. The palm of the hand then faces in the direction of the magnetic force that acts on a positive charge. ight hand f the moving charge is negative, the direction of the force is opposite to that predicted by H-1. Notes: (1) F is always perpendicular to both B and v. () This rule works even when v is not exactly perpendicular to B. (3) Details of this H is different from that adopted in Essential Physics but it gives the same result. F v B 13 The next two examples make use of the following equations in the equation sheet: Note: To do these problems, you need to be able to do the following derivation based on these equations: mv F qvb r mv r qb Below are the trajectories of three charged objects with the same mass and the same magnitude charge moving through a region of uniform magnetic field. ank the objects based on their speeds > > 3. > 1 > > > > 1 > 5. Cannot tell v 15. Below are the trajectories of three charged objects with the same mass and the same magnitude charge moving through a region of uniform magnetic field. The magnetic field is pointing out of the page. Which of the following are correct about the sign of the particles charge? 1. Only 1 is negative.. Only is negative. 3. Only 3 is negative. 4. Only and 3 are negative. 5. All are negative. 6. Cannot tell v 16 The force on a current-carrying wire A magnetic field exerts a force F qvbsin on a single moving charge, so it's not surprising that it exerts a force on a current-carrying wire, seen as containing a set of moving charges. Using q = t, the force is: F vtbsin n the above, the charge, q, moves through the wire in time t. So, vt = L, the length of the wire and The magnetic field from a long straight wire The long straight current-carrying wire, for magnetism, is analogous to the point charge for electric fields. The magnetic field at a distance r from a wire with current is: 0 B r F LBsin The direction of the force is given by H-1, where your thumb points in the direction of the current. Current is defined to be the direction of flow of positive charges, so your right hand always gives the correct direction. 17 0, the permeability of free space, is: Tm/A Figure above shows the field lines about a current-carrying straight wire. 18 3

4 The magnetic field from a long straight wire The field from a circular current loop The direction of the field follows another right-hand rule: At the center of the circular loop, the magnetic field is: ight-hand ule No.. (H-) Curl the fingers of the right hand into the shape of a half-circle. Point the thumb in the direction of the conventional current, and the tips of the fingers will point in the direction of the magnetic field. 19 o B where is the radius of the loop. The above shows that the field lines around the loop resemble those from a bar magnet. The direction of the magnetic field can be determined by yet another H as shown at right. 0 The field from a solenoid The force between two wires N turns L t is straightforward to show that the force on wire one due to the field, B 1, produced by the current in wire causes a magnet force in wire one, F 1 that has the same magnitude but opposite direction to that of F 1. B 1 B 1 For a solenoid of length L, current, and total number of turns N, the magnetic field inside the solenoid is given by: N 0 B L The direction of the magnetic field is determined by the same H that determines the direction of the magnetic field in a current loop. 1 F 1 1 F 1 The force between two wires n this situation, opposites repel and likes attract! Parallel currents going the same direction attract. f they are in opposite directions they repel. The next four examples make use of the following circled equations in the equation sheet: Make sure that you can see why B is pointing up at where wire is! 3 4 4

5 5. The figure shows two current-carrying wires separated by a distance of 1 mm and the same length of 30 cm. Further, 1 = 1A and = A. Which of the following statements are correct about the forces between the wires? 1. The forces can be calculated and they are attractive.. The forces can be calculated and they are repulsive. 3. The forces are attractive, but there is insufficient information to tell the magnitude of the forces. 4. The forces are repulsive, but there is insufficient information to tell the magnitude of the forces The figure shows two current-carrying wires with 1 = 1 A and = A. Which of the following statements is correct about the forces between the wires? 1. The force on wire is bigger than that on wire 1.. The forces on wire 1 and are the same n which direction is the net magnetic field at the origin in the situation shown below? All the wires are the same distance from the origin. 1. Left. ight 3. Up 4. Down 5. nto the page 6. Out of the page 7. The net field is zero 8. A solenoid is 10 cm long and contains 100 turns. What is the B field inside the solenoid when a current of 1 A is applied? (Use 0 = 4 x 10-7 N/A ) T T T T T T Hint: Fill out the following for the fields B i at the origin due to individual currents i : Direction of B 1 is: Direction of B is: down Direction of B 3 is: Direction of B 4 is: up The ranking of the B i s is: B 1 > B = B 4 > B Magnetic Flux Magnetic flux is a measure of the number of magnetic field lines passing through an area, A. The equation for magnetic flux,, is: = BAcos where B is the magnetic field and is the angle between the B vector and the area vector, A. The S unit for magnetic flux is the weber (Wb). 1 Wb = 1 Tm A 60 o A 90 o = 0 o = 60 o = 90 o Maximum Zero Figure. llustration of how the magnetic flux through a coil varies with the angle. 9 A Faraday s Law Faraday's Law says that the emf induced in a coil of N turns is given by the rate of change of the magnetic flux in the coil: Faraday's Law: N t Since = BAcos, the magnetic flux changes if any of B, A or changes. Faraday s law has important applications in the generation and distribution of electricity. We call the voltage induced by a changing magnetic flux an induced emf. n other words, changing magnetic flux acts like a battery in a coil or loop, and a current flows when there's a complete circuit. 30 5

6 Lenz s Law Faraday s law tells you that there will be an induced emf in a circuit if the magnetic flux it is linking changes. Faraday s law also tells you the magnitude of the induced emf. But Lenz's Law tells you the direction of the emf: Lenz's Law: A changing magnetic flux induces an emf that produces a current which sets up a magnetic field that tends to oppose whatever produced the change. Coils and loops do not like any change in the magnetic flux they are linking, and they will try to produce an emf or current to counteract the change imposed on them. Note that the net result is that they usually would not be successful - the change in the magnetic flux still takes place. This tendency to oppose the change is why there is a minus sign in Faraday's Law. 31 A pictorial approach to Lenz s Law Example: A wire loop in the plane of the page is in a uniform magnetic field directed into the page. Over some time interval the field is doubled. What direction is the induced current in the loop while the field is changing? Step 1: Draw a Before picture, showing the field passing through the loop before the change takes place. Step : Draw an After picture, showing the field passing through the loop after the change. Step 3: Draw a To Oppose picture, showing the direction of the field the loop creates to oppose the change. Step 4: Use the right-hand rule to determine which way the induced current goes in the loop to create that field. 3 The next example make use of the following circled equations in the equation sheet: The next 4 examples make use of the pictorial method A wire loop is located near a long straight current-carrying wire. f the current in the wire is decreased, 1. The induced current in the loop would be clockwise.. The induced current in the loop would be counter-clockwise. 3. There will be no induced current in the loop. 10. The loop is now placed directly on the wire with the wire bisecting the loop. f the current in the wire is increasing, in what direction is the induced current in the loop? 1. The induced current is clockwise.. The induced current is counter-clockwise. 3. There is no induced current

7 11. f the rod (in red) is moved to the right, will there be an induced current? f so, in what direction is it? 1. Clockwise. Counterclockwise 3. There is no induced current 8. f the right edge of the rectangular circuit loop is steadily moved to the right, what is the direction of current induced in the loop, for an observer looking from the north pole? Observer on this side. N S 1. Clockwise. Counter-clockwise Describing the motion at ALL points of a wave So, for a wave traveling right, if we say the motion of the particle at x = 0 is given by: y(0, t) Asin( t) The motion of a particle at another x-value is: y(x,t) = Asin(t - kx) (ight-traveling wave) where k is a constant known as the wave number. Note: this k is not the spring constant. This one equation describes the whole wave. Similarly, for a wave traveling left, the motion of the particle at position x is: y(x,t) = Asin(t + kx) (Left-traveling wave) 39 The next three questions make use of these equations and concepts: Which of the following determines the wave speed of a wave on a string? 1. the frequency at which the end of the string is shaken up and down. the coupling between neighboring parts of the string, as measured by the tension in the string 3. the mass of each little piece of string, as characterized by the mass per unit length of the string. 4. Both 1 and 5. Both 1 and 3 6. Both and 3 7. All three. 41 Making use of the mathematical description 1 1 yxt (, ) (0.9 cm)sin (5.0 s ) t (1. m ) x Which of the following is the closest to the wave's wavelength? m. 1. m m 4. 5 m m 4 7

8 Making use of the mathematical description 1 1 yxt (, ) (0.9 cm)sin (5.0 s ) t (1. m ) x Which of the following is the closest to the wave's frequency in Hz? Hz. 0.8 Hz Hz 4. 5 Hz Hz Making use of the mathematical description 1 1 yxt (, ) (0.9 cm)sin (5.0 s ) t (1. m ) x Which of the following is the closest to the wave's speed? m/s m/s m/s m/s 5. 4 m/s The Doppler Effect in general n some situations, both the source and the observer move. The general Doppler equation combines the previous results. 18a. Let's say you, the observer, move toward a stationary source with speed v o. For you, what change in the wave? f v v o f v vs This, exact same equation is given in the equation sheet A. Their speed relative to you B. Their wavelength C. Their frequency Use the upper signs when the source and observer are approaching each other, and the lower signs when they are moving away from each another. 1. A and B only.. A and C only. 3. B and C only. 4. All When the source doesn t move, there is no change in the wave pattern. So, the wavelength remains the same. But the apparent speed of the wave is changed to v + v o if the source is approaching an observer or to v - v o if the source is leaving an observer. The new frequency is the apparent wave speed divided by the wavelength. 18. You, a stationary observer, are near a source that is moving with speed v s. What change, in effect, for the waves as seen by you? A. Their speed relative to you B. Their wavelength C. Their frequency 1. A and B only.. A and C only. 3. B and C only. 4. All

9 Standing waves When the source moves, an observer being approached sees the wave fronts compressed, while an observed being moved away from sees the wave fronts dilated. The new frequency is the wave speed divided by the new wavelength. The two waves can be represented by the equations: y 1 = A sin(kx - ωt) and y = A sin(kx + ωt) The resultant wave is their sum, and can be written as: y = A sin(kx) cos(ωt) Simulation This is quite different from the equation of a traveling wave because the spatial part is separated from the time part. t tells us that there are certain positions where the amplitude is always zero - these points are called nodes. There are other points halfway between the nodes where the amplitude is maximum - these are the anti-nodes Standing Waves 1. The figure at right shows the pattern of a standing wave on a string with one end fixed. What is the wavelength of this wave in terms of the length of the string, L? 1. L/3. 3L 3. 4L/5 4. 5L/4 L

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