PHYS 2211L LAB 6. The Current Balance

Transcription

1 PHYS 2211L LAB 6 The Current Balance Purpose In this experiment we will measure the magnetic force between two current carrying wires. From our data, we will calculate a value for the permeability of free space (µ 0 ). Principles A basic fact in electrodynamics is that charges in motion generate magnetic fields. An electric current through a wire will create a circular magnetic field that wraps around the wire in a direction given by the right hand rule: if you point the thumb of your right hand in the direction of the current, your fingers will curl in the direction of the magnetic field generated (see Diagram 1). Current 1 Magnetic Field Lines Diagram 1: Magnetic field lines generated by a current-carrying wire. Another fact is that magnetic fields exert forces on moving charges. If an electric current passes through a magnetic field, the charges that make up the current will feel a force whose direction is given by a second right-hand rule: if you point your right index finger in the direction of the current and your middle finger in the direction of the magnetic field, your thumb will point in the direction of the magnetic force that the charges feel (see Diagram 2). Make sure you distinguish between these two phenomena. A current (moving charges) generates a magnetic field which may exert a force on other moving charges (another current). The field does not affect the charges that generate it. On the other hand, charges that do feel this force will set up their own magnetic field (since they are in motion). This second field will then exert a force on the first set of charges. In this way, two currents will exert forces on each other. Currents that are parallel with each other will exert an attractive force on each other. Currents that are anti-parallel will 75

2 PHYS 2211L LAB 6 The Current Balance exert a repulsive force on each other. We will measure these forces using the current balance. Force Thumb Middle finger Field lines Inde x Current 2 Diagram 2: Force on a current in a magnetic field. Also: the right-hand rule. The Current Balance The current balance is designed to measure the magnetic forces between currents. It is fitted with a balanced array of thin metal rods that serve as conducting wires. When connected to a power source, current travels through the rods in a continuous path. One arm of the circuit is balanced on knife-edge supports by counterweights. The other arm is fixed in place. The path is arranged so that the upper arm passes above and parallel to the fixed arm. By adjusting the counterweights and balancing the system with known masses, we can measure the magnitude of the magnetic forces between them. Magnitudes of the fields and forces The magnitude of the magnetic field around a current-carrying wire can be derived from Amperes Law. The result is (1) I1 B = µ 0 2 πd 76

3 PHYS 2211L LAB 6 The Current Balance where B is the magnetic field strength, µ 0 is the permeability of free space and d is the distance from the center of the wire. B is measured in Tesla (newtons per amperemeters) and µ 0 has the value 4π x 10-7 Tesla-meter/Ampere. Here, I 1 is the current that creates the field. The magnitude of the force on a segment of a current carrying wire that passes through a magnetic field is given by (2) F = I 2 Bsinθ Here, I 2 is the current in the segment, l is the length of the segment, B is the field strength and θ is the angle between the current and the field. For our experiment we will let I 1 be the current in the bottom (fixed) wire and I 2 be the current in the top wire. The magnitude of these currents will be equal, and we will call their common value I. Also, the angle between the field and the current will be 90 degrees, so that sin θ = 1. Putting this all together, we have (3) F = µ 0 2 πd I 2 for the force on the top wire exerted by the bottom wire. The force will be attractive when the currents are in the same direction and repulsive when the currents are opposite. We will wire the circuit for opposite currents and a repulsive force. See Diagram 3. The experimental procedure will be to first balance the system at a given separation distance with the current turned off. We will use an optical lever setup to determine equilibrium, as we did in the Coulomb Balance experiment. We will place a small mass on the movable wire and adjust the current until the system balances. The repulsive force between the wires will then be equal to the weight of the mass. I 2 d Diagram 3: Top & bottom wire segments with anti-parallel currents. The magnetic field B created by I 1 is out of the page above the bottom wire and into the page below it. I 1 77

4 PHYS 2211L LAB 6 The Current Balance The Earth s Magnetic Field A complication in the experiment is that the Earth s magnetic field will exert a small but measurable force on the wires. This field has a strength of about 5 x 10-5 T the same order of magnitude as the fields generated by the wires. Also, magnetic fields generated by the electrical wiring in the room may affect the experiment. We can address these ambient fields by taking two steps. First, we take advantage of the fact the force exerted by a magnetic field depends on the relative directions of the field lines and the current (see equation 2 above). The Earth s field at Atlanta runs very close to due North-South and is angled slightly above the horizontal. The vertical component of the combined ambient field will not affect our experiment (why?). The horizontal component can be rendered negligible by orienting the wires in the same direction as the field. Then, with sinθ 0, the effect of the ambient field will be close to zero. The second step is to take two measurements of the current for each weight, with the current reversed for the second measurement. The current directions in the two wires will be opposite in both cases, and the force on the upper wire from the lower will always be upward. But the force exerted by the ambient field will be upward in one case and downward in the other. Averaging the two current values will be a good approximation to the balancing current in the absence of any external fields. 78

5 The setup is illustrated in the diagram below. The experimental procedure will be to first balance the two current-carrying rods at a given separation distance, with the current turned off. We will use an optical lever setup to keep track of this equilibrium position, as we did in the Coulomb Balance experiment. We then place a small mass on the upper rod. By adjusting the current, we can balance the weight of the mass with an upward magnetic force. To account for the effect of Earth s magnetic field, we take two measurements for each mass placed on the upper rod: one for current running one way through the rods and another for current running in the reverse direction. Averaging the two current values will be a good approximation to the balancing current in the absence of any external fields. Equipment Current Balance 2 Low Voltage Power Supplies Digital Multimeter DPDT switch Milligram mass set Laser Micrometer Ruler 20 gram mass (spacer ~ 3.6 mm) 50 gram mass (as a weight) Spirit level Magnetic dip needles Banana wires 1. Adjust and align the current balance. Level the base of the balance using the leveling screws and a spirit level. Place the knife-edge balance arm on the two support posts and make sure the balance arm can rotate freely, with the damping blade fitting into the slot in the base of the balance. Adjust the alignment of the two front rods so that they are horizontal and parallel to each other: -The upper rod can be adjusted forward and back on either end. -The lower rod can be adjusted up and down on either end. 79

6 Place a mass on the upper rod so that the rod is pressed against the lower rod. Try to eliminate any gaps between the rods then they are placed together. Note: You should have 3 masses in your experiment kit. The 10- and 20-gram masses are to be used as spacers in the experiment. The 50-gram mass is to be used as a weight (not a spacer). Use the magnetic dip needle to determine the direction of the ambient magnetic field at your experiment station. (The direction will vary somewhat at different locations within the lab.) Place the current balance so that the front conducting rods are parallel with the horizontal component of the ambient field. 2. Set up the circuit. Construct the circuit in the diagram below using the current balance, 2 low voltage power supplies and a DMM set to ammeter (A): A Current Balance Wired for repulsive force (anti-parallel currents) 2 LVPS Upper & lower rods Diagram: The Current Balance The power supplies are connected in parallel. This will give the highest current. Keep the power off until ready to take measurements, but turn the current knobs on both supplies to full now. Turn the voltage knobs to their lowest setting. Connect the DMM for currents in the range 0-20 amps. Connect in series with the lead from the power supplies. Select the high range current setting. 80

7 Connect the leads from the power supply to the banana plugs on the balance, as described below. To reverse the current through the balance during the experiment, simply switch the positive and negative leads. Note: A DPDT (double-pole double-throw) switch can be used to reverse the current polarity without having to change the connections of the leads. The positive and negative leads from the power supplies should be connected to the two poles on one end of the switch. The two leads from the current balance should be connected to the two middle poles. When the switch is thrown one way, the current runs forward through the balance; it is reversed when the switch is thrown the other way. When the switch is in the vertical position, the circuit is broken and the current is off. The current balance should be connected so that there is a continuous path for current through the two conducting rods. For a repulsive force between the rods, the path should run in one direction through the upper rod and in the other direction through the lower rod. Note that the metal posts that support the fixed and movable rods are part of the conducting path. Also, current runs through the knife-edges which support the movable arm. Connect one lead from the power supplies (or DPDT switch) to one knife-edge support post. Current will run up through this post, through the knife-edge, through the movable rod, around and back to the other knife-edge and down the other support post. Connect this support post with a banana wire to a fixed-rod support post at the front of the device. For a repulsive force (anti-parallel currents), connect to the corresponding support post at the front of the balance. For an attractive force (parallel currents), connect to the front support post that is diagonal from the back post. Finally, connect the last post to the other power supply connection. To reverse the current during the experiment, simply switch the leads from the power supplies. 4. Set up the optical lever. Set the thinner spacer (~ 3.5 mm) on the lower rod and another mass as weight on the upper rod. The upper rod should be pressed against the spacer. Place the laser in front of the apparatus. Plug it in and turn it on. Shine the laser on the mirror on the movable arm of the balance. Set up a screen or use the wall to catch the reflection from the laser. Tape a strip of masking tape under the laser spot and draw a horizontal line through its center. The line will mark the equilibrium position in the experiment. 81

8 Note: The laser can damage your eyes. Do not look into the laser beam or its reflection. Take care not to point it at anyone s face. Remove the weight and spacer. Adjust the counterweights so that the laser spot returns to the equilibrium position. 5. Take current measurements. Measure and record the current necessary to balance the system when a small (milligram range) mass is placed on the upper rod. When the system is balanced, the laser spot will be at the equilibrium position. Take measurements for masses ranging from 10 mg to 100 mg in 10 mg increments: Turn on the power supplies. Both current knobs should be on full. Start with one voltage knob at the lowest setting (counterclockwise). Use the other voltage knob to control the current. For the highest currents (12-14 amps), turn up both voltage knobs. Place a mass in the pan on the upper rod. Using the voltage knob on one power supply, increase the current until the laser spot returns to the equilibrium position. Reduce the voltage while you record the value of the balancing current. Reverse the connections to the current balance (or reverse the DPDT switch) so that the current runs in the other direction through the rods. Increase the voltage until you find the current value for the reversed current. Reduce the voltage while you record the balancing current. Turn off the power supplies, the laser and the DMM when finished. 6. Take a second data set. Repeat the above procedures using the other spacer provided. Set up the optical lever to mark this new equilibrium position, then determine the forward and reverse currents necessary to balance the system when weights are placed on the upper rod. Alternatively, use an attractive force for the second data set: Switch the connection to the power supplies so that the current through the rods is parallel, giving an attractive magnetic force. With the current off, place a mass on the upper rod and adjust the counterweights until the rod returns to the equilibrium position. 82

9 Take the mass off the upper rod. Because of the counterweights, the rod will swing up. Increase the current until the magnetic force pulls the rod back to equilibrium. Record the current. Reverse the current and find the current necessary to balance the system in this case. Repeat for masses in 10 mg increments from 10 to 100 mg. 7. Parameter Measurements Measure the diameters of both conducting rods on the current balance with the micrometer. Measure the thickness of the mass spacers with the micrometer. Take three measurements for each of these and use the average values in your calculations. Take the diameter readings at different places along the rods. Measure the length of the top front rod with a ruler. The length you need runs from the center of one connecting (perpendicular) rod to the center of the other. Measure to the nearest tenth of a millimeter. Analysis For each data set: Calculate the weights in newtons of each of the masses. Calculate the center-to-center distance between the rods for each equilibrium spacing. This is the sum of the thickness of the spacer and the radii of the two rods. Calculate the average of the forward and reverse currents for each weight. Calculate the square of the average current for each weight. 2 Graph the magnetic force (equal to the weight) as a function of I average. You may use the DataStudio file FBC.ds on the lab computers. Determine the slope of the graph and write down the equation of the graph. Calculate µ 0 from the slope. See equation (3) above. Average your two values for µ 0 and take the percent error from the accepted value: 4π x 10-7 Tesla-meter/Ampere. 83

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