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Magnetic Fields – Tangent Galvanometer Introduction and Theory: Just like an electric field exists around electric charges, there is a magnetic field surrounding a permanent magnet and around moving electric charges. Since electric current is a flow of charge, there is a magnetic field around any current carrying wire. This magnetic field can be detected by observing the behavior of a compass needle in the presence of current carrying elements. Like an electric field, the magnetic field also is a vector quantity and has both a magnitude and a direction. The direction of a magnetic field at any point in space is the direction indicated by the north pole of a small compass needle placed at that point. The magnetic field of the earth is thought to be caused by convection currents in the outer core of the earth working in concert with the rotation of the earth. The field has a shape very similar to the field produced by a bar magnet. Incidentally, the north magnetic pole of the earth does not coincide with the north geographic pole. In fact, the north magnetic pole is located close to the Earth's South Pole (in Antarctica), while the south magnetic pole is located close to the Earth's North Pole (in Canada). For a loop of wire consisting of N turns wound close together to form a flat coil with a single radius R, the magnetic field resembles the pattern of a short bar magnet, and its magnitude at the center of the coil according the Biot-Savart law is () (1) where is the permeability of free space (4π × 10-7 T·m/A) and I is the current in the coil. If the current is expressed in amperes (A), and the radius in meters (m), the unit of magnetic field strength is Tesla (T). Note that this field vector is parallel to the axis of the coil. In many situations the magnetic field has a value considerably less than one Tesla. For example, the strength of the magnetic field near the earth’s surface is approximately 5∙10-5 T. The more convenient unit of magnetic field strength is a gauss (1 G = 10-4 T). The instrument used in this experiment is a tangent galvanometer that consists of 1-5 turns of wire oriented in a vertical plane that produce a horizontal magnetic field. The direction of the magnetic field at the center of the wire loop can be determined with the help of the right-hand-rule. If the curled fingers of the right hand are pointed in the direction of the current the thumb, placed at the center of the loop, indicates the direction of the magnetic field. The magnetic field of the coil is parallel to the coil axis. Figure 1 shows the vector sum Bnet of the Earth's magnetic field (BEarth) and the magnetic field due to the current (BLoop) for the case when the coils of the galvanometer are oriented so that the Earth's magnetic field (BEarth) is parallel to the plane of the coils. The magnetic field due to the current (BLoop) being perpendicular to the coils plane will then be perpendicular to the Earth's field. Therefore the relationship between the horizontal component of the earth's magnetic field BEarth and the magnetic field of the coil BLoop can be expressed as tanθ = BLoop / BEarth (2) where θ is the angle between BEarth and Bnet. From equations (1) and (2) we get () (3) This can be rewritten as tan =M·N·I (4) where ( ) = constant. The horizontal component of the earth's field can now be found by measuring the field due to the coils and the direction of the net magnetic field relative to the direction of the earth's field. The angle θ can be found by using a compass. If the compass is first (with no current: I = 0) aligned with the magnetic field BEarth and then current is supplied to the coils, the compass needle will undergo an angular deflection θ. Because of the relationship given by equation (4) this equipment is called a tangent galvanometer. Note that for θ = 45o, tanθ = 1 and BLoop = BEarth. Objectives: BEarth Bnet B θ Loop Figure 1. Vector sum of the magnetic fields. To verify:  the vector nature of magnetic fields;  that the field at the center of a current loop is normal to the loop and directed in accordance with right hand rule; To investigate the relationship between the magnetic field and:  the number of turns - B(N);  the value of the current - B(I) inside a current carrying coil. To determine the strength of the horizontal component of the Earth’s magnetic field. Equipment: Virtual Tangent Galvanometer with two views: Overhead and Oblique. Virtual DC power supply, ammeter and compass mounted in the center from the Tangent Galvanometer Apparatus lab Procedure: Before starting the experiment please get practice with the virtual equipment! Log in to Virtual Physics Labs using your KET ID and password. Load the virtual “Tangent Galvanometer Apparatus Lab” and familiarize yourself with the setup. The apparatus is viewed from two perspectives: Overhead (Figure 2a), and Oblique (Figure 2b). You will switch between views using the buttons at the top left edge of the screens. Take some time to become familiar with each view. In the Overhead view shown in Figure 2a, you see two vector arrows. One represents the horizontal component of the Earth’s magnetic field. The other represents the magnetic field produced by the current-carrying wire loops. Neither vector automatically points in the appropriate direction. Rather these vectors can be rotated as needed by dragging the points of the arrows. The entire apparatus can be rotated in the overhead view by dragging the Handle. The coil unit has a compass mounted in the middle. With no current applied to the coil, the compass responds only to the horizontal component of the earth’s magnetic field. Figure 2a. View 1: Overhead Figure 2b. View 2: Oblique The Oblique view shown in Figure 2b does not rotate. Explore the following in the Oblique view. A frame with a pair of vertical supports provides two nails which hold 1 to 5 circular loops of insulated wire. A horizontal platform holds a sheet of polar graph paper for measuring angles in the horizontal plane. The compass at the bottom right provides a close - up of the real compass. You will take compass reading there. Remember that the red end of the compass is its north end (seeking Earth's North Pole). Notice how the deflection of the compass is affected by the power switch, the voltage adjust knob, and the number of loops of wire. When the power is on and current flows through the loop, a magnetic field due to the current is produced inside the loop. We expect it to be normal to the plane of the loop. If the Earth’s magnetic field were nonexistent the compass needle would point in the direction perpendicular to the loop’s plane. However, under the influence of the two magnetic fields, the compass takes the direction of their resultant field Bnet. The two views are completely independent. You will only work with one view while performing a given part of the lab. You will use the overhead view for part 1 and the oblique view in part 2 and part 3 of the lab. Part 1. The direction of the magnetic field at the center of a current loop You will use the overhead view for this part of experiment to verify that the magnetic field of the current loop’s BLoop at the center of a loop is normal to the plane of the loop. In this view the number of loops N is fixed at 4 and the current I is fixed at 3.0 A when the power is turned on. Begin with the power turned off. Drag the handle to orient the frame so that the 0° end of the loop is pointing north - the direction of the red end of the compass. Drag the end of the vector BEarth to point in the magnetic north direction as shown in Figure 3a. Now with the current off, the needle points in the same direction as the BEarth and as you turn on the power supply the needle will deflect showing the direction of the net magnetic field Bnet which is vector sum of the fields BEarth and BLoop. You might want to arrange the BLoop vector to point in the direction you think is correct. You will need to experiment a bit. It will involve switching the current on and off and rotating the apparatus. Rotate the apparatus and find the position(s) where the needle stays still when the power is turned on and off. You should have noticed that there are two different orientations of the loop that result in no change in the needle’s direction when the current is turned on and off (this is because the magnitude of the BEarth is greater than BLoop as you can see from Overhead view). At these two orientations of the loop the direction of the total magnetic field Bnet (and therefore the direction of the needle) is unchanged (only the magnitude is changed) when the current is turned on and off. In other words, in these orientations of the loop the magnetic field vectors BEarth and BLoop are parallel or antiparallel (see Figure 4a and Figure 4b). Figure 4a. Figure 4b. Hopefully, you have noticed that as the loop is at right angles to the Earth’s field the needle does not move when the current is turned on and off. So the loop’s field is perpendicular to the plane of the loop with two possible directions - 180° apart. But only one of them can match our right hand rule. Align again the direction of the vector BEarth with compass needle (to North) as in Figure 3a. You have confirmed that the loop’s field BLoop is perpendicular to the loop - up (East) or down (West). You also observed that with the current on, the compass always points in the direction of Bnet - about 37° east of north. Because Bnet is in the second quadrant so it must have a north and an east component. BEarth supplies the northward component, so BLoop must be to the east as in Figure 3b. In your lab report show (qualitatively) the vector addition of BEarth with each of the two possible BLoop. What is the current direction? Is the current flowing into the screen at 180° (and out at 0°) or into the screen at 0° (and out at 180°)? Apply the right hand rule to figure out the current direction. Part 2. The magnetic field at the center of a current loop You will investigate the relationship between the strength of the magnetic field at the center of loop and: a) the number N of loops; b) the current I through the loop. Equation (3) shows that the field BLoop at the center of the loop is directly proportional to the tangent of θ (the Earth’s field remaining constant). Therefore, you should find that the plots of tanθ vs. E N Figure 3a. Figure 3b. Bnet BLoop BEarth BEarth BLoop Bnet N or tanθ vs. I both should yield a straight line through the origin (equation (4)). You will use the oblique view this time and investigate these relationships to test equation (1). To read the compass as accurately as possible use zooming with right-click on the apparatus and select “Zoom In” from the menu. You can then drag the apparatus around as needed. To test the effect of the number of turns N on the strength of the loop’s field BLoop, measure the angle of deflection (with respect to north) of the compass for 1 to 5 loops by keeping the current at constant value I = 3 A. Enter your data in Logger Pro and plot both θ vs. N and tanθ vs. N on the same graph using “Right Y-Axis” feature: on toolbar select Options > Graph Options > Axes Options > mark Right Y-Axis (be sure the preferences in LP for angles are set in degrees: select File > Settings for startup > Degrees). Are both graphs linear? Describe the graph tanθ vs.N. Apply linear fit to the graph tanθ vs. N to find the slope with uncertainty and assuming that the radius of the circular loop R=20 cm, calculate the value of BEarth,N with the error. To test the effect of the current through the loop on the strength of the loop’s magnetic field, measure the angle of deflection for currents of 0 to 3.5 A in 0.5 A increments keeping the number of the loops fixed at N = 5. In Logger Pro plot both θ vs. I and tanθ vs. I on the same graph using “Right Y- Axis” feature from Graph Options described above. Are both graphs linear? Describe the graph tanθ vs. I. Apply linear fit to the graph tanθ vs. I to find the slope with uncertainty and calculate the value of BEarth, I with the error assuming R=20 cm. Compute the average value BEarth = (BEarth,N + BEarth, I)/2 with the error. Part 3. The strength of the horizontal component of the Earth’s magnetic field According to equation (1) the field at the center of the loop is directly proportional to the product N·I. Now you will test the complete equation by using it to calculate the horizontal component of the Earth’s magnetic field BEarth and compare it with the value from part 2. You will do that as follows. Using a convenient point on the line of best fit from near the middle of your graph tanθ vs. I find the value of the current and by equation (1) calculate BLoop for that value of the current (hint: do your calculation for the current when tanθ = 1). Assume that the radius R=20 cm for circular loop. Compute the horizontal component of Earth’s magnetic field BEarth using equation (2) for given value of tanθ and calculated value of BLoop (notice if tanθ = 1, then BEarth = BLoop). Compare your calculated value of the earth’s magnetic field Bearth with the average value from part 2. Final conclusion: Do your experimental findings support equation (1)? How does the magnetic field of a coil depend on the current in the coil? * Include answers to all questions in lab report Magnetic Field (Tangent Galvanometer) - Post lab checklist Part 1. The direction of the magnetic field at the center of a current loop o Show (qualitatively) the vector addition of BEarth with each of the two possible BLoop. o Answer on the questions. Part 2. The magnetic field at the center of a current loop o Measure the angle of deflection (with respect to north) of the compass for 1 to 5 loops by keeping the current at constant value I = 3 A. o Plot two graphs: θ vs. N and tanθ vs. N on the same graph. o Describe the graphs. o Using the slope of the graph tanθ vs. N calculate BEarth,N with the uncertainty (USING ERROR PROPAGATION). o Measure the angle of deflection for currents of 0 to 3.5 A in 0.5 A increments keeping the number of the loops fixed at N = 5. o Pro plot both θ vs. I and tanθ vs. I on the same graph. o Describe the graphs. o Using the slope of the graph tanθ vs. I calculate BEarth,I with the uncertainty (USING ERROR PROPAGATION). o Compute the average value BEarth = (BEarth,N + BEarth, I)/2 with the percent difference between the two values. Part 3. The strength of the horizontal component of the Earth’s magnetic field o For given value of tanθ =1 calculated value of BLoop, and then compute the horizontal component of Earth’s magnetic field BEarth using equation (2).

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