PHYS272_Lab4

=Physics for Scientists and Engineers- Lab 4 PreLab =  Assignment: Read chapter 3 of the book Experimentation: An Introduction to Measurement Theory and Experiment Design, paying particular attention to section 3-10. Perform the calculations described below. This assignment is due at the start of the laboratory period. These concepts will be used to complete the lab.

Questions:
 1. Suppose that you take 5 measurements of magnetic field using a gaussometer and obtain the following results: 0.184 T, 0.201 T, 0.179 T, 0.163 T, and 0.177 T. Calculate the mean of your measuremnts and the standard deviation of the mean using equation s 3-6 and 3-7 from Experimentation. If we assume that the uncertainty in the measurement comes primarily from variations in different readings and not from inaccuracies in the gaussmeter, the standard deviation of the mean can be used as the uncertainty in the magnetic field measurement. Report the mean magnetic field and uncertainty.  2. You take measurements of the radius and different lengths of a solenoid and find that R = 2.0 ± 0.1 cm, L 1 = 10.4 ± 0.1 cm and L 2 = 12.3 ± 0.1 cm. If θ 1 = tan -1 (R/L 1 ) and θ 2 = tan -1 (R/L 2 ) calculate θ 1 and θ 2 as well as their uncertainties. Then if N = 100 ± 1 and the number of turns per unit length n = N/(L 1 + L 2 ) find n and the uncertainty in n. Finally if I = 2.0 ± 0.5 A, μ 0 = 4π × 10 -7 NA -2 and B = (1/2)μ0 nI(cosθ2 + cosθ1 ) calculate the magnetic field B and its uncertainty. Use the expression for calculating the uncertainties given in Prelab 3.  3. You take a measurement of the radius of a coil of wire R = 5.2 ± 0.1 cm and the distance from the coil's center z = 3.4 ± 0.1 cm. If f = (z 2 + R 2 ) -3/2 calculate f and the uncertainty in f.

=Physics for Scientists and Engineers - Lab 4 =

Objectives:

 * To determine the forces on a current carrying coil due to the magnetic field.
 * To determine magnetic field strengths generated by simple current geometries

Equipment:

 * Power Supplies, Pasco, 8 amp
 * Table clamp, long rod, short rod, 90 degree clamps, test tube clamp
 * Bell Gauss meters with Hall element probes
 * Banana leads (3) and alligator clips (2)
 * Magnetic Force Coils
 * Force sensor with 500 g mass for calibration
 * Digital Multimeter
 * Solenoid with side slit,
 * Long wire mounted on a meter stick
 * Short cm ruler
 * Round coil
 * Data Studio and Graphical Analysis software

Physical Principles:
**Magnetic Force** The force on a current-carrying wire segment in a magnetic field is given by, math F= BIl sin \theta math where B is the magnetic field (in units of Tesla), I is the current, l is the length of the wire, and θ is the angle between the current and the direction of the magnetic field. For current flowing through multiple wires, such as a rectangular coil, the force on a single wire must be multiplied by the number of wires in the coil, N. If B and I are perpendicular to each other, then sin θ = 1, and the equation simplifies to, math F= NBIl math  According to the Biot-Savart Law, the strength of the magnetic field at a point P, caused by the current I in a short segment of wire of length L is given by the following formula: math \displaystyle B=\frac{\mu_0Il sin \theta}{4\pi r^2} math In this equation, the angle θ is measured between the direction of the current flow and the direction of the point P from the current segment and r is the distance to the point. B is perpendicular to the plane of I and r in the direction of advance of a right hand threaded bolt when rotated in the direction of I into r through the angle θ (not more than 180º). The standard unit of measurement for magnetic field (B) is Tesla (T). Another popular unit for this parameter is Gauss (G) which is given by: 1 Tesla = 10 4 Gauss. In equation (3), the constant μ 0 is given by 4π X 10 -7 Tm/A. For the following simplified geometries, the Biot-Savart Law (equation 3) may be written in a simplified form.  The magnetic field lines produced by a long straight wire are circles concentric with the current. In this geometry, the Biot-Savart Law simplifies to, math \displaystyle B=\frac{\mu_0I}{2\pi r} math The right hand rule can be used to obtain the direction of the magnetic field. If the right hand thumb points in the direction of current flow, the right hand fingers of a half-closed hand point in the direction of the magnetic field. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> The magnetic field inside a long solenoid with n = N/L turns per unit length carrying a current I, is given by math B= \mu_0 n I math <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> For solenoids that are not very long, the following formula is used to find the magnetic field: math B= \frac{1}{2} \mu_0 n I (cos\theta_2 + cos\theta_1) math The angles θ 1 and θ 2 are those subtended by the radius R of the ends as observed at the point where the magnetic field is measured. They are given by math \displaystyle \theta_2=tan^{-1}({\frac{R}{L_2}}) \, \mathrm{and} \, \theta_1=tan^{-1}({\frac{R}{L_1}}) math where L 1 and L 2 are the distances from the probe to the ends of the solenoid. See figure 1. The direction of the magnetic field is parallel to the solenoid axis. With the fingers of the right hand wrapped around the solenoid in the direction of the current flow, the right hand thumb points in the direction of B inside the solenoid. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> The magnetic field on the axis of a circular coil with a radius R and observed at a distance z along the axis from the plane of the coil is given by math \displaystyle B=\frac{\mu_0 N I R^2}{2(z^2+R^2)^{3/2}} math With the fingers of the right hand wrapped in the direction of current flow, the right hand thumb points along the axis in the direction of the magnetic field.
 * Magnetic Field due to an Electric Current**
 * 1. Long straight wire**
 * 2. Solenoid**
 * 3. Circular Coil**

=<span style="color: #333333; font-family: Arial,sans-serif; font-size: 2em;">Procedure: =

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Magnetic Force
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">Hang the rectangular coil from the force sensor and attach the ends of the wires of the coil to the power supply with the banana clips and wires. The bottom edge of the coil should be centered between the poles of the permanent magnet. Measure the number of turns, N, of the coil, and the average length, l of the bottom section of the coil, and record these values in your lab book. Using a Bell Hall element gaussmeter, measure the magnetic field at five equally spaced points along the bottom length of the coil. Record the average of these five values as the average magnetic field. Calculate the standard deviation of the mean of the measurements as you did in your prelab and record it as the uncertainty in the magnetic field. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Open the PASCO Capstone software and calibrate the force sensor, setting the force due to the equilibrium weight of the coil equal to zero. Hang a mass from the coil to set the maximum weight. Set the stop time to 1 second, and the sample rate to 100 Hz. Turn the Pasco power supply on, with the current set at 0A as measured by the multimeter, take the first data set, recording the mean and standard deviation of the force. Repeat for values of +1 A, +2 A, +3 A, +4A, and +5 A. Reverse the leads on the power supply to take data at -1 A, -2 A, -3 A, -4 A, and -5 A. Record the uncertainty in each of your current readings using the information [|here]. <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Open Graphical Analysis and make a graph of force vs. current including error bars. Fit the data to a straight line and record the value of the slope. Right click on the box showing the slope and add the slope error. According to equation for the magnetic field from a long straight wire, the slope will be equal to NBl. Calculate NBl based on your gaussmeter measurements of magnetic field etc along with the uncertainty. Does the calculated value of NBl with its uncertainty overlap with the slope?

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Magnetic Fields of simple current configurations
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">Since it is time consuming to set up the apparatus for the magnetic field experiments, you will take turns at stations already prepared. Below are several things that you should remember throughout the experiment.
 * Every time the Bell magnetic field meters are turned on, they have to be recalibrated. The instructions to do this are on the back of the device.
 * Both the voltage and current knobs on the power supply should be set to maximum.
 * Do not leave the power supplies on for too long when they are connected to the wires since this causes the wires to overheat.
 * Uncertainties in magnetic field measurements can be calculated from [|here].

=== Spring 2016 Instructions: Sadly, the department's very sensitive Gaussmeter is being repaired. Please skip the following Straight Wire Portion <span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;"> ===

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Straight Wire
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;"> Secure a 1.5 m lead along the length of a meter stick. Connect the positive lead from the power supply across one end of the straight wire. Connect the lead from the negative terminal to the common receptacle of the multimeter. Connect a banana lead into the receptacle marked 20 A (or 10 A). Connect the other end of this lead into an alligator clip and connect the clip to the other end of the long straight wire. Orient the meter stick horizontally along the north-south direction. Set the current to maximum (8 amps). Measure the magnetic field at a distance r of about 5 mm from the center of the wire. The flat face of the probe should be vertical and in the plane of the wire (see figure 2). //Turn the current off as soon as you are done measuring the magnetic field.//Measure and record in the data recording section the distance r from the wire in meters, the current I in amps and the magnetic field B in tesla.

Compare the measured value with the value calculated from the equation for the magnetic field from a long straight wire by finding the uncertainty in your calculation.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Solenoid
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">Orient the solenoid with its axis horizontally in the east-west direction. Connect the positive lead from the Pasco power supply across one end of the solenoid. Connect the lead from the negative terminal to the common receptacle of the multimeter. Connect a banana lead into the recepticle marked 20 A (or 10 A). Connect the other end of this lead into an alligator clip and connect the clip to the other end of the solenoid. Be certain that the enamel has been removed from the ends of the wire. Set the current to 8 amps. Insert the Hall element gauss meter probe into the slit on the side of the solenoid until the probe end is on the solenoid axis and measure the magnetic field. //Turn the current off as soon as you are done measuring the magnetic field.// Measure and record in the data recording section the radius R of the solenoid, number of turns N, the distances of the Hall probe from the ends L 1 and L 2, the number of turns per unit length n = N / (L 1 + L 2 ), the current I in amps and the magnetic field B in Tesla. Compare the measured value with the calculated value for the magnetic field from the equation for a solenoid. For greater accuracy, use the equation for a short solenoid to calculate the magnetic field B, using math \theta_1=\tan^{-1}(R/L_1) \, \mathrm{and} \, \theta_2=\tan^{-1}(R/L_2). math <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">Does the equation for the short solenoid improve the error between calculated and measured magnetic fields? Repeat your measurement and calculation for the magnetic field at the end of the solenoid using the hole at the end of the solenoid.

<span style="color: #333333; font-family: Arial,sans-serif; font-size: 1.125em;">Circular Coil
<span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">  <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">Mount the coil coaxially on a meter stick with the meter stick along the coil axis in the east-west direction (see figure 3). Mount the Bell Hall element probe of the gauss meter on the coil axis with the flat face of the probe vertical and perpendicular to the coil axis. Set the current to about 8 amps. For a series of distances z of the probe from the plane of the coil, measure the magnetic field. Use at least five values of z between about 4 cm and 14 cm. Using Graphical Analysis plot a graph of B vs. math 1/(z^2+R^2)^{3/2}, math <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">including error bars, and compute a value of μ 0 from the value of the slope which is math \displaystyle \frac{\mu_0 N I R^2}{2} math <span style="color: #333333; display: block; font-family: Arial,sans-serif; font-size: 14px;">as predicted by the equation of the magnetic field from a circular coil.