Experiment 2-3. What s Happening Between Currents? -Lorenz Force-

Transcription

1 Experiment 2-3. What s Happening Between Currents? -Lorenz Force- Purpose of Experiment A current-carrying electric wire produces a magnetic field. When a closed current-carrying wire is placed in a magnetic field, it feels a torque. If the magnetic field is not uniform (that is, the magnitude and direction depend on the position) such as near a straight wire, two wires exert forces on each other. In this experiment, we measure the torque acting on an electric wire which is placed in a magnetic field produced by a solenoid with dc current. We determine the direction of the torque and see the dependence of the magnitude of the torque on the magnetic field and the current flowing in the wire. - From E. Hecht, Physics An electric motor would be a typical example of practical use of the torque that a current (or currentcarrying wire) experiences in a magnetic field. But an amperemeter (or voltmeter) we frequently use in the laboratory also employs the torque in uniform magnetic field as left picture. And a speaker also employs the torque acting on the coil in nonuniform magnetic field as right picture. What would be the direction of the magnetic field? Pay attention to the position of speaker's coil.

2 Outline of Experiment Confirm a closed current-carrying electric wire experiences a torque in a uniform magnetic field. Examine the characteristics of a magnetic torque exerting on a wire - How does it depend on the direction and intensity of the magnetic field due to the current flowing through the solenoid? - What changes do you make if you change the current direction and intensity of the closed circuit? - How does the magnetic field created by the current in the closed circuit affect? In addition, understand the principles and theoretical calculations of the current balance used in this experiment - How to use DC power supply Fig.1 Schematic illustration of a set of DC dual power supply and current balance sets 1. If you connect the alligator clip and the wire like Fig.1, the current direction is determined as Fig.1

3 Good Bad Fig.2 Good example and bad example of current balance weight before connecting wire Fig.2 As you can see in fig.2, the parallel weights, which depend on the current balance, should be adjusted so that they can be paralleled when there is no magnetic field, before current is connected by connecting the wires Fig.3 Actual appearance of power supply and how to use dial 3. I will explain how to use it simply through the real image of Fig.3. If the equipment is well prepared through steps 1 and 2, 1 Turn on main power. -> 2 Activate the output terminal with the on / off button. -> 3 Use the P1 / P2 buttons to move the cursor to each P1 and P2 item on the panel. -> 4 The V / I button allows you to move the cursor to the voltage and current sections. -> By using the left and right buttons of 5, it is possible to adjust by moving the cursor from the 10th digit to the 2nd decimal place in the voltage and current section.

4 4. In the case of the power supply unit, the maximum current value is set in the current section. Therefore, the maximum current value can be set by the desired current, and the voltage may be increased to the maximum value. That is, it is set to be adjusted by using the voltage. Rather, you can record the voltage and current values during the experiment and experiment with the voltage. 5. Next is the actual operation of the current circuit diagram and the wire connections described in Fig.1. (a)-1 (b)-1 (a)-2 (b)-2 (b)-3

5 (a)-3 Fig.4 In case of (a)-1 ~ 3, the positive electrode of the current balance is connected to the negative electrode of the power supply. In this case, the positive voltage is across the current balance. On the other hand, in case of (b)- 1~ 3, the anode of the current balance is connected to the com pole of the power supply. Negative voltage is applied as in Fig.1 6. As described in 5, understanding how the power supply and circuit diagram work together can help you understand how the current balance is affected by the magnetic field of the solenoid. In Fig.4(a), the force acts down as described in Fig.1 On the other hand, Fig.4(b), the force acts up. Experimental Method These equipments are prepared in the laboratory. (Parentheses mean the number of them.) - solenoid (500 turns, 1) - dc dual power supply (1) - electric wire with crocodile clip (6) - variable resistor (rheostat) (1) - current balance (1) - 30 cm ruler (1) - digital scale (1, common use) - micrometer (2, common use) - Scotch Tape (2, common use)

6 - cutter knife (2, common use) - If you need more stuff, inquire to your teaching assistant or experiment preparation room (19-114), or prepare yourself. Learn in advance about the current balance. The following is a recommended experiment method # Since the voltage and current are allowed to flow in the power supply, the option is used in this experiment. You can use 1A or 2A as needed.# 1) Connect the solenoid to a port of the dual power supply. [Video1] 2) Make sure that the current control knob of the power supply is turned counterclockwise, and turn on the power supply switch. 3) Adjust the weight adjusting screw so that the balance is slightly tilted up or down to maintain the balance without touching the solenoid's wall. 4) Move the current control knob of the power supply connected to the balance so that the ammeter indicates 1 A. [Video2] 5) When the power supply is switched on and off, check that the balance moves in the opposite

7 direction from the original tilted position, and reverse the polarity of the current applied to the solenoid or balance if it is in the same direction. 6) Changing the nine kinds of current value of the current balance between 0~2A, and record the value, that pointer indicates. 7) Also, repeat the measurement of 6) for 5 values for the current in solenoid in 0 ~ 2.5 A. (At this time, it is possible to keep the current flowing in the balance constant by changing the wire with the solenoid and the crocodile gripper inserted in the balance.) [caution 1 : The currents in balance and in solenoid should not over 2 A and 2.5 A respectively. Because a current above 2 A could heat the wire, don't flow it overlong.] [caution 2 : Don't detach the balance from the support screw while a current flows. (The spark could cause some damage.) [caution 3 : Make sure that the balance does not touch the inner wall of the solenoid.] 8) How does the current in the magnetic field depend on the current flowing in the conductor? How do you rely on your yard? Can we compare the measurement results quantitatively with theoretical calculations? It is recommended to write experiment notes in the following way. (example form of experiment form Turn density of solenoid m = /m Width of current balance d = m Length from center line of balance to copper(wire) l = Length from center line of balance to the tip of indicator s = Mass of balance m = kg Radius of support axis of balance r = m Current in solenoid I = A m m Current in balance I'(A) Position of ruler y(m) Displacement of indicator y(m)

8 Slope y/i' = m/a Current in solenoid Slope y/i'(m/a) Slope ( y/i')/i = m/a 2 Background Theory A force between currents in separated wires can be understood as one current feels a magnetic field induced by another current. Let the currents in parallel wires be ia and ib respectively and the distance between them d. The magnitude of magnetic field induced by ia is B a = μ 0i a 2πd (1) at the position current ib flows and the direction is downward by the right hand rule. μ0 is the permeability which is defined as μ0 = 4π 10-7 T m/a in vacuum or matter with no magnetism. It will

9 be explained later again. The force exerted on length L of a wire b with current ib is F ba = i b L B a (2) According to the picture, the magnetic field and the current ib is mutually perpendicular, so that the magnitude of the magnetic field is F ba = i b LB = μ 0 i a i b L/(2πd) (3) and the force per unit length is F ba /L = μ 0 i a i b /(2πd) (4) Since the force is measurable, for two wires carrying the same current i separated by 1m, we define 1 A (ampere) as the magnitude of the current i which will produce the force of N. Then the proportionality constant in equation (4) (permeability) is μ0 = 4π 10-7 N/A 2 (5) or using 1 N = 1 A m T (6), it becomes μ0 = 4π 10-7 T m/a (7) When a loop carrying current I is inclined at φ in a uniform magnetic field such as in an ideal solenoid, it experiences a torque by the magnetic field.

10 In the picture, for horizontal length a, vertical length b and turn number N=1, the magnitude of the torque is given by τm = IabBsinφ (8) In an ideal solenoid, the magnetic field is parallel to the axis of solenoid and the magnitude is given by B solenoid = μ 0 i solenoid n (9) where n is the turn per unit length, namely turn density of solenoid. In reality, since the length of a solenoid is finite, the magnetic field exists outside the solenoid and the inside field is also slightly different from ideal case. For a solenoid of length l and radius r, the magnetic field at the center is B solenoid = μ 0 i solenoid n cosθ c (10) where θc is the angle between the axis and the straight line from the center of solenoid to the end. In this case, cosθ c = l 2 {r2 + ( l )2 } (11) When a loop is partially included in the magnetic field like this experiment, the torque acting on the loop(current I) by the magnetic field B solenoid is τ m = ( 1 ) μ 4 0Iabi solenoid nl {r 2 + ( l )2 } sinφ (12) The current balance uses the equilibrium between the magnetic torque above and the restoring torque

11 exerted by the balance. The force exerted by the magnetic field is not only the case of the torque acting on a current-carrying wire. In fact, there were discussions about whether a magnetic force acts on a wire or a current itself. For now, we understand it as charges which carry the current feel the force first and then almost simultaneously the wire feels it since they can't get out of it. When a particle with the charge q moves at the velocity v in a magnetic field B, this particle experiences the magnetic force of F = qv B (13) This is called Lorentz force. With an electric field E, Lorentz force denotes, including the electric force, F = q E + q v B (14) The first example of Lorentz force is the discovery of electron by J. J. Thomson, which is still used to control the direction of electron beams in Braun tube in computer monitor (CRT, Cathode Ray Tube) From E. Hecht, Physics In cathode ray tube (CRT), to deflect the electron beams one can use the electric field (left picture) or the magnetic field. The method using the magnetic field which is employed in home televisions and computer monitors uses the yoke coil (right picture) instead of the deflection electrode. If it has tiny pixel size like a natural color television, very tiny magnetic field from Earth or nearby magnetic

12 substances could affect on its color system. Indeed, the exported televisions, depending on Earth's magnetic field at their destination, are corrected in advance. Things to Figure Out Current Balance A current balance uses the mechanical equilibrium condition between the magnetic torque a wire feels and the torque by gravitational force. The current balance used in this experiment is an electric circuit board coated by copper, which can rotate around its center axis. Note that the support axis which also does a role of the electrode wire is attached on the upper side of the circuit. It gives the stability so that the balance can do its roles. Now, we will see how a current balance keeps up the stability. Assume the balance is in equilibrium when no current flows. When the solenoid applies a magnetic field B, the balance feels a torque τm, so it starts to rotate. But for the balance where the support is attached on the upper side, when the support is rotated, the perpendicular line passing circuit's center of mass doesn't pass the contact point of the

13 support. Thus the restoring torque by gravity acts on the circuit. Let the mass of the balance be M, the radius of support be r and the gravitational acceleration be g. Then the restoring torque is τ = Mgrsinθ (A1), so that the torque increases as the angle θ increases. When it compensates for the magnetic torque, the circuit becomes balanced. In the equilibrium, the relation between the torque and the angle is given by τ m = Mgrsinθ Mgrθ (A2) If the angle is small, they are mutually proportional. In this experiment, we calculate the angle from the displacement of indicator attached on the balance. Let the distance from the support to the indicator be L and the vertical displacement of the indicator be y. Then the angle of circuit is θ = tan 1 ( y L ) y L (A3) and the torque acting on the circuit is τ m Mgr y L (A4) Note that we have assumed the circuit is negligibly thin and uniform and it becomes horizontal in the equilibrium state with no magnetic field. References Treatment of measurement data Analysis method based on the graph Sir Joseph John Thomson An excellent trainer who found the electron Robert Millikan Pride of modern American physics The Exploratorium Science Snackbook - Motor Effect

14 Current balance