Ampere's Law. Introduction. times the current enclosed in that loop: Ampere's Law states that the line integral of B and dl over a closed path is 0


 Alyson Jackson
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1 1 Ampere's Law Purpose: To investigate Ampere's Law by measuring how magnetic field varies over a closed path; to examine how magnetic field depends upon current. Apparatus: Solenoid and path integral board, DC power supply with builtin voltmeter/ammeter, cables, Hall Effect magnetic sensor, meter stick Introduction Ampere's Law states that the line integral of B and dl over a closed path is 0 times the current enclosed in that loop: B dl = 0 I enclosed You have seen the usefulness of the law in determining, without complicated integration, the magnetic field merely by knowing the currents enclosed by a path with a high degree of symmetry, such as a circular loop around a long straight wire. For such a wire, the magnetic field is given by: B long straight wire = 0 I 2 R
2 2 (For a long straight wire carrying current I the B field line direction is tangent to centered circles. Along those circles, B is constant, so we can pull it out of the integral. The remaining integral is simply the circumference of the circle, which is 2 R, so B long straight wire 2 R = 0 I ) In your experiment, I must be understood as NI multimeter since the single wire of the solenoid contributes I to the enclosed current with each winding (turn) of the solenoid) : B dl = 0 N I = 0 N I multimeter in general.equation 1 If there is no net current within the closed path, the closed integral is zero. This does not necessarily mean there is no B field present along the line integral, or no currents enclosed. Rather it means that the dot product with the field direction sums to zero. Note that Up and down currents through the enclosed surface must be assigned opposite signs. Think of two adjacent wires with equal and opposite currents. The closed line integral surrounding them is zero. If the closed line integral is not zero, you know that there is a net current within the closed path which is generating a magnetic field. In this lab you will actually sum up the contributions of B dl over such a path around a solenoid, to check if their sum does indeed equal 0 times the current enclosed by your path. The magnetic field around the solenoid will be determined by a magnetic sensor; you will measure the output voltage of the sensor (which is proportional to B) using the Fluke multimeter. The current through the solenoid will be measured by the readout on the power supply. SOLENOIDS An application of Ampere s Law involves a solenoid (a wire coil wound on a cylinder) with: N = number of turns of solenoid (dimensionless) R = radius of coil (meters) I multimeter = current through solenoid (amperes) L = length of solenoid (meters). The B field intensity at the center of the solenoid is calculated to be: B C = 0 N I ammeter 4R 2 L 2 Equation 2 At the end of the solenoid it is: B E = 0 N I ammeter 2 R 2 L 2 Equation 3.
3 N can be determined from B (since can measure all other parameters) at either of these points, as well as from the line integral of tangential B along any loop passing through the coil. 3 HALL EFFECT A magnetic field can be measured with a Hall Effect sensor. In the diagram below, a current, I, is transmitted through a silicon semiconductor. The potential between the top and bottom points is zero until a perpendicular magnetic field is applied which exerts a force on the moving charges. If the current consists of positive charged carriers, a positive charge will accumulate at the lower end of the conductor. Negative carrier flowing in the same direction as I will induce a negative charge at the lower end. Thus, the Hall Effect can distinguish the charge of the carrier! (In this figure, the convention that current is a flow of positive carriers is shown.) In any case, a small but measurable potential is induced by the magnetic field. If the field is reversed, so is the polarity of the induced voltage. You will use a Hall Effect magnetic field detector to measure the magnetic field. The Figure 1: Hall Effect detector is mounted at one end of a clear plastic block that can be oriented in a magnetic field. The output is amplified and recorded by a sensitive voltmeter. Evaluation of the line integral of the Bfield s parallel components for two or more closed paths through a solenoid will determine the number of solenoid turns N, by application of Equation 1. In evaluation of the line integral we will approximate infinitesimal line elements dl by finite elements l : 0 N I ammeter = B dl B l cos = B l l for every contribution is simply the distance between the line segment markings. The angle in the dot product is always 0 degrees, since we are always orienting the black line along the path, so the above equations reduces to: 0 N I ammeter = B dl B l cos = B l Do not reverse the direction of circulation around the loop during your summation.
4 Keep magnetic material (steel watch bands, bracelets, etc.) away from the experiment. Besides the solenoid, currents in power supplies, computers, etc. produce magnetic fields 4 Procedure A. Calibration In a recent lab you compared a handheld magnet's magnetic field to the that of the Earth's (known to be 0.3 G or 0.3 x 104 T in Piscataway). Here you will calibrate the Hall sensor by using the Earth s magnetic field so that we can convert the Hall voltage (V) to a magnetic field (T or G). Note that the Hall sensor measures the component of any magnetic field along the black line inscribed on the clear plastic sensor holder. Since we do not know the direction of the horizontal field of the earth, the sensor will be rotated 360 degrees to determine the effective zero reading for the sensor. The sensor will register the lowest voltage (magnetic field) when it is oriented towards the Earth's North Magnetic Pole and the highest voltage (magnetic field) when it is pointed towards the Earth's South Magnetic Pole, which should be 180 degrees from North. At angles in between them the voltage will register some intermediate value, but never zero  this is the offset voltage from the circuit that amplifies the true Hall voltage. You will use the high and low voltage readings to: a) Find the baseline (offset) voltage in the Hall circuit b) Find the conversion factor k to go from Volts > Tesla 1. Make sure the power supply, which supplies a current to the solenoid is turned off in this part. Turn on the Fluke multimeter and set it to measure voltage  this will read the Hall sensor voltage which is proportional to magnetic field. Voltage should be zero before the power supply is turned on, since the power supply not only supplies the solenoid current; it also powers the Hall sensor. The voltmeter should register a voltage around 2.4 V. 2. Place the Hall sensor, mounted on the clear plastic holder, on the ZERO FIELD COMPASS flat on the path integral board with the black cable. Rotate the holder and watch the voltage vary  it will read a maximum in the direction of the Earth's Magnetic South and a minimum in the direction of Earth's Magnetic North. Remember that magnetic field lines go from the North to the South on a bar magnet. Which direction is Earth Geographic North (golf course, ARC building, lecture hall, Davidson)? Write your answer in the spreadsheet. (Hint: the Earth's Magnetic South corresponds to its Geographic North.) 3. Take voltage data as you rotate the sensor holder through 360 degrees, from 12 o'clock to 11 o'clock. See figures below. Record and calculate conversion factor k in handin sheet.
5 5 B. Proportionality check of B vs. I How to set up the power supply for Variable Current (Constant Voltage) Turn the Fine and Coarse knobs on the Voltage side of the power supply to their maximum clockwise positions. Set the Volts/Amps selector switch (near the LED display) to Amps, then adjust the Fine and Coarse knobs on the Current side of the power supply so until the desired current value reads out on the display.
6 6 1. Turn on the power supply and adjust the knobs as described above; this will allow a current to flow through the solenoid. Place the Hall sensor holder inside the center of the solenoid, oriented along the its length (9 o'clock compass direction). See figure below. 2. Keeping the Hall sensor fixed and motionless, measure V as a function of I as you increase the current from 0.04 A to 0.18A in 0.02 A increments. Switch the power supply (variable side) leads and record from A to A in 0.02 A increments. Record your data in handin sheet. 3. Open Logger Pro (go to Desktop > Lab Softwares > Logger Pro 3). Enter your data and plot B vs. I. Do a linear curve fit. Look on the graph to identify the B value on the fit line where the current I=0. V 0 =B 0 /k will be the new offset value you subtract from any further measurements. It is determined from more data points (16 vs. 2 in Part A), so it should yield a more accurate value of the offset. Enter this value in your handin sheet. Print graph. Close Logger Pro. C. Closed line integral enclosing zero net current (Loop A)
7 7 1. Open up Ampere.GA3, which is a Logger Pro file in the same folder as this writeup. You will enter your voltage data here ( Va raw data, Vb raw data, Vc raw data ) and later convert it to B field and B l values that you can plot and sum. 2. Set the power supply so that it puts out 0.15 A to the solenoid coil. You will maintain this solenoid current for the remainder of the lab. Place the Hall sensor holder on Loop A (no current enclosed whatsoever), putting the tip (the sensor itself) right at the labeled starting point. Follow the path along the arrows indicated, taking voltage readings at each 1 or 2centimeter line segment marking. Make sure the black line on the sensor holder is oriented along the path at every point. Record B and 'Line segment #' for each segment of the path. Remember not to measure the start position twice. 3. After you finish taking data for the loop, enter your values for V 0 and k in the Logger Pro data table under their headings. You may wish to fill down with these repetitive values by doubleclicking on the column heading, check the Generate Values box and select Numerical Fill, then entering the values of V 0 and k. 4. Doubleclick on 'B dl' column heading to see incomplete column definition, for example: "B(Loop A)"
8 8 Complete 'B dl' column definition by multiplying B data by your delta L (0.01 m or 0.02 m). For example: "B(Loop A)"* Repeat Step 4 for columns B dl (B) and B dl (C).. Bring the Graph to the front and examine the graphs.. Use the AnalyzeIntegral function to sum up the B dl contributions ( B dl B l ). Write this value down in the handin sheet and print the graph. D. Closed line integral enclosing nonzero net current (Loop B) 1. Repeat Part C above for Loop B (nonzero net current enclosed). 2. From Equation 1, calculate the number of turns N in the coil. Record in the handin sheet. E. Closed line integral enclosing zero net current (Loop C) Repeat Part C again, this time for Loop C (zero net current enclosed). F. Comparison of end and center solenoid field strengths 1. Measure the length and radius of the solenoid using the meter stick. 2. Measure the magnetic field, with the sensor holder oriented along the solenoid axis (lengthwise), at the center and end of the solenoid (these are clearly marked on the part of the path inside the solenoid. Keep in mind that the sensor is at the tip of the sensor holder.
9 9 3. Combine Equations 2 and 3 to find an expression for the ratio of the field strengths at the end of the solenoid to that of the center of the solenoid B end B center. Write this in your handin sheet, showing all your equation manipulations. Compare this to your measured values, and again write in handin sheet. 4. Calculate the number of turn N, from your data. Record in handin sheet and compare with the calculation of N from Part D.
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