Lab 5: Measuring Magnetic Field of Earth

Transcription

1 Dr. W. Pezzaglia Physics B, Spring 010 Page 1 Las Positas College Lab 5: Magnetic Field 010Mar01 Lab 5: Measuring Magnetic Field of Earth Mar 1, Monday: Lab 5 (today) Lab # due Video: Mechanical Universe # Magnetism Discuss: Graphing techniques using Ecel, slope, uncertainty in slope Mar 8, Monday: Lab 6 (net week) Activity: Ampere s Law Based on: ================================================================== Part I. Setup Start with butcher paper on your table, have a compass and protractor. A. The Earth s Magnetic Field 1. The magnetic field in our lab room is very disturbed by all the metal beams in the walls and floors. You will need to find a place in the room where the field points in the same direction over the size of the top of your table ( feet). You may need to move your table quite a bit! Question 1: Setup What is your lab station number? Make a sketch and show the eact position in the room your lab table is at (with measurements to the wall).. Orientation: Draw on your paper the direction of Building North, True North and the Local Magnetic Meridian. The definitions are: Building North is the direction that the building is facing (from your point of view, towards the blackboard, running parallel to the walls) True North is the direction of the north pole of the earth. It is approimately clockwise from building north. Local Magnetic Meridian is the direction that the compass points. Magnetic Deviation is the angle (measured clockwise) of the magnetic north relative to the true north. The Livermore Airport lists it as 16 east, but what we measure in the room will be modified by the metal beams. Question : Summary of Orientation a) What is your approimate magnetic deviation? b) How does it compare with the epected value (from airport)? c) How much does the local magnetic meridian deviate from building north?

2 Dr. W. Pezzaglia Physics B, Spring 010 Page Las Positas College Lab 5: Magnetic Field 010Mar01 B. The Magnet Some of the magnets might be bad. Make sure you have a good one, i.e. that it is a true dipole. Sometimes during storage they end up too close to another magnet and they end up with a third pole (see figure). Also, magnets sometimes reverse their polarity, so you need to make sure that north is really north. Recall the definition is that the north pole of a magnet seeks the earth s north pole (i.e. points north). You can dangle the magnetic on a string to test this! Also make sure that your compass needle is pointing north rather than south! Question : Eplain how you have tested your magnet to make sure it is good. In particular, note if the poles are reversed from the labeling! ======================================================================= Part II. Field Mapping We shall use the field of your magnet fighting the field of the earth to determine the relative strength of one to the other. A. Neutral Point Method 1. Aligned Align magnet along the local meridian. The field to the left(right) of the magnet is in the opposite direction to the Earths, and drops of like the cube of the distance. Find the point at which the field is eactly cancelled by the earth using a compass or sensor. Mark on paper, and call this distance 1. Find point on the other side of magnetic:. Note the strength of the magnetic field at distance from the magnet of dipole strength m is: 0 m B m = ( ) π Question : Report results (a) Are the points 1 and symmetric as epected? (b) For each point, calculate the implied ratio of the magnetic dipole to the earth s field: m π = Be 0 (c) What are the SI units of this ratio?

3 Dr. W. Pezzaglia Physics B, Spring 010 Page Las Positas College Lab 5: Magnetic Field 010Mar01. Second Neutral Point Method Align magnet OPPOSITE the local meridian. The field along the z ais of the magnet now opposes the earth s field, but drops off like the cube of the distance Find the point north of the magnet at which the field is eactly cancelled by the earth using a compass or sensor. Mark on paper, and call this distance z 1. Find point on the south side of magnetic: z. Note the strength of the magnetic field at distance from the magnet of dipole strength m is: 0 m B m = π z Question 5: Report results (a) Are the points z 1 and z symmetric as epected? (b) For each point, calculate the implied ratio of the magnetic dipole to m z π the earth s field: = Be 0 Question 6: Summary of Results (use tabular format!) (a) Compare your measured values of the ratio (m/b e ). (b) Calculate the average value (c) Calculate the standard deviation (use STDEV function in Ecel) Stdev (d) Calculate the absolute uncertainty: σ =, where N= for four measurements. N (e) What is the PERCENT uncertainty in your value of the ratio (m/b e )? B. Deviation Method In the above method it is difficult to determine the neutral point. Instead we put the magnet perpendicular to the earth s field and measure the angle between the two fields. 1. Put magnet perpendicular to earth s field. Put compass some distance from magnet. Measure the deviation angle θ of the compass (from magnetic north).. Do for several values of above (try to get deviations in the range from 80 down to 10 ) 5. Do for several values of below (call them negative, and the angles also negative) 6. Theory (vector addition of perpendicular vectors) says Bm 0 m 1 tanθ = = Be B π e 1 Hence a plot of tan(θ) vs would have an epected slope of: 0 π m B e

4 Dr. W. Pezzaglia Physics B, Spring 010 Page Las Positas College Lab 5: Magnetic Field 010Mar01 7. Report data in tabular format, (see sample below) make the plot using Ecel, and determine the slope and intercept, and most important, the uncertainty in the slope (see sample below) Use the Ecel template posted on the web site to create your plot. In particular, it will determine the best line through your data. The R value tells you the goodness of the line (i.e. if its close to 1 its good, if its 0 its terrible). Sample Data: Deviation of Field from Magnet X(m) Deviation Theta (radians) Tan(theta) 1/(^) Stdev Slope STEYX Count 9 Unc Slope %unc.% Magnetic Deviation y = R = Tan(deviation) cubed inverse distance (meters)

5 Dr. W. Pezzaglia Physics B, Spring 010 Page 5 Las Positas College Lab 5: Magnetic Field 010Mar01 Question 7: Summary of Results (a) Is your plot linear as epected? (b) Is the intercept zero as epected? If not, interpret. (c) What is the slope, and its uncertainty? (be sure to include proper units) (d) From slope, determine the value of (m/b e ). Its percent uncertainty is the percent uncertainty of the slope. C. Second Deviation Method Same idea, but this time measure along the perpendicular ais. 1. Measure the deviation as a function of distance z.. Repeat measurements for the other side (call them negative z, and deviations also should be negative). 1. The analysis is similar, ecept there is a factor of in the equations: A plot of tan( θ ) 1 vs should be a line, and the slope will again yield a measurement of z. Again, make data table, plot, determine slope and intercept. Question 8: Summary of Results (a) Is your plot linear as epected? (b) Is the intercept zero as epected? If not, interpret. (c) What is the slope, and its uncertainty? (be sure to include proper units) (d) From slope, determine the value of (m/b e ). Its percent uncertainty is the percent uncertainty of the slope. 0 π Question 9: Comparison of Results (a) Compare the results for the two methods of deviations. Are they consistent? Does one method behave better than the other? (b) Compare your results for the deviation method with the earlier neutral point method. Are they consistent? Which method do you think is best (why)? ======================================================================= Some Notes: If the plots are not linear then we may have to look at the fact that the dipoles are not point-like 0 mz but have physical size (length L ). The correct formula is: B( z) =. Hence we π ( z L ) 1 z would have to plot tan( θ ) vs. For field perpendicular its: ( z L ) 0 B( ) = π m ( + L ) /, so plot tan(θ ) vs 1 ( + L ) / m B e.

6 Dr. W. Pezzaglia Physics B, Spring 010 Page 6 Las Positas College Lab 5: Magnetic Field 010Mar01 Part III. Oscillation of Magnet in Earth s Field By measuring the period of oscillation of the magnet in the earth s field we obtain a measure of the product (mb e ). Combined with results from part II we can then individually measure the strength of the earth s field and the strength of the magnet. A. Moment of Inertia 1. Measure mass of magnet M in kg.. Measure dimensions a, b, L.. Calculate the moment of inertia for the flat configuration. Be sure you use kg and meters! 1 I f = M [ L + b ] 1. Calculate the moment of inertia for the on edge configuration (be sure to use kg and meters!) 1 I e = M [ L + a ] 1 Question 10: Summary of Results (a) Sketch magnet and summarize the parameters: M, a, b, L. (b) Report the moment of inertia calculations. (c) Compare: how much do the two moments of inertia differ? Does the orientation matter? (d) Do we really have to worry about the dimensions a and b or are they so small that we 1 can just use the formula: I 0 = ML? 1 B. Measurement of Period 1. Setup: Be sure to do the oscillation eperiment in the eact location that you made your measurements for part II. Dangle magnet on a string (use a stand?). Be sure to record which orientation you used, flat or on edge.. Measure period of oscillation. Suggest you measure time elapsed for a number of cycles to get a more accurate result.. Repeat your measurement to 5 times to get consistent results. Record data in tabular format!. Compute the average period, its standard deviation, and its uncertainty Question 11: Summary of Results (a) What is the average period? (b) What is the standard deviation? (c) What is the (absolute) uncertainty in your period measurement? [The uncertainty in the Stdev average of N items is:: σ = ] N (d) Convert to a percent uncertainty

7 Dr. W. Pezzaglia Physics B, Spring 010 Page 7 Las Positas College Lab 5: Magnetic Field 010Mar01 Part IV: Analysis: Measure m and B Question 1: Oscillation of Magnet: Theory says the period is given by: T = π (a) From your measurements of period T and calculated moment of inertia, etract the value for the quantity (mb). (b) The percent uncertainty in this quantity will be half the percent uncertainty in your period. (c) What are the units of the quantity (mb)? Question 1: Etracting m and B. From Part II you have a measurement of the ratio of the magnetic moment to the m magnetic field of the earth. Lets call this ratio R : R = B e From Part III you have a measurement of the product. Lets call it Q : m R = Oscillation of Magnet: Theory says the period is given by: Q = ( mbe ). B e (a) From these two values, compute the magnetic moment: m = RQ. Be sure to have the correct units! Q (b) Net compute the earth s magnetic field strength: B e =. Be sure to have the correct R units! (c) The percent uncertainty of both of these quantities will be given approimately as half the sum of the percent uncertainty in R and Q. I mb Question 1: Magnetic Field of Earth (a) Compare your measured value of magnetic field of the earth to the epected value of approimately Tesla. (b) If available, use a magnetic probe to get the actual value of the magnetic field in the room and compare to your measured value. Question 15: Magnet (a) Calculate the pole strength q m of your magnet, which is related to the magnetic moment m by the simple formula: m = L qm (b) What are the units of the pole strength?

8 Dr. W. Pezzaglia Physics B, Spring 010 Page 8 Las Positas College Lab 5: Magnetic Field 010Mar01 V. Notes on Ecel and Uncertainties To get slope and intercept in Ecel: Slope Function: m=slope(y values, values) Intercept Function b=intercept(y values, values) R Squared Test Value R =RSQ(y values, values) Uncertainty in Slope: Where N= number of points Where Standard Error in Y values: Where Standard Deviation of X values: σ = m σ δy N δy STEYX(y values, values) σ STDEV( values) Uncertainty in Intercept: Where N= number of points Where Standard Error in Y values: Where Standard Deviation of X values: Where Average of X values: δy σ b = N 1+ σ δy STEYX(y values, values) σ STDEV( values) =AVERAGE( values) Propagation of Uncertainties: Recall absolute uncertainty in addition or subtraction is a sum in quadrature of the absolute uncertainties. For eample, if Z=X+Y, then the absolute uncertainty in Z is given by the formula: σ = σ + σ z y Recall percent uncertainty in multiplication or division is a sum in quadrature of the percent uncertainties. For eample, if Z=XY, then if we call e the percent uncertainty in : e z σ = (100%), then the percent uncertainty in Z is given: X z e = e + e Recall percent uncertainty in a N-th power, is N times the percent uncertainty. Hence if you take the square root of a quantity the percent uncertainty is decreased by a factor of. y