Magnetic Force and the Determination of μ₀

Magnetic Force and the Determination of μ₀#

Danger

⚠️ ⚠️ ⚠️ ⚠️ WARNING ⚠️ ⚠️ ⚠️ ⚠️

⚠️ ⚠️ ⚠️ ⚠️ HIGH CURRENT IN USE TODAY ⚠️ ⚠️ ⚠️ ⚠️

⚠️ ⚠️ ⚠️ ⚠️ DO NOT TOUCH THE METAL CONDUCTORS (BARS) ⚠️ ⚠️ ⚠️

⚠️ ⚠️ ⚠️ ⚠️ YOU CAN G- Reflect on this week’s lab; what did you learn?USLY INJURED ⚠️ ⚠️ ⚠️ ⚠️

⚠️ ⚠️ ⚠️ ⚠️ WARNING ⚠️ ⚠️ ⚠️ ⚠️

Background#

Overall goals and overview#

  • Experimentally determine the permeability of free space \(\mu_0\)

  • Along with your previous measurement of \(\varepsilon_0\), determine the speed of light \(c\)

  • Understand the relationship between magnetic force and current using a consistent induced magnetic field, reasonably produced by high current through parallel, cylindrical conductors

  • Conduct three rounds of nine trials each (total 27 trials) of applying more mass (more gravitational force) and more current to produce a magnetic field (and therefore more magnetic force). Do the rounds sequentially rather than all of a single applied mass in a row to ensure you catch and correct any significant errors early (e.g. if the apparatus is bumped out of alignment) rather than having to do all 27 trials again. The order of the trials for applied masses would be 0, 25, 50, 75, 100, 125, 150, 175, 200, 0, 25, 50, 75, 100, 125, 150, 175, 200, 0, 25, 50, 75, 100, 125, 150, 175, 200 mg

  • In this experiment, first determine the separation distance, \(d\). Then add mass with a series of increasing masses and apply the necessary current \(I\) applied along the conductors so they return to parallel where the separation distance returns to \(d\). Under this condition, the magnetic force required to maintain the separation \(d\) will be the difference of the gravitational force on the applied mass. Then determine \(\mu_0\) for each trial, and an overall value through both averaging of all your trials and plotting of all your data (i.e. magnetic force vs. current² between the conductors). Comparing then your results from both analysis methods to the accepted \(\mu_{0\text{-accepted}}\). Ultimately determine the speed of light \(c\).

By measuring the magnetic force between two parallel current-carrying conductors, the permeability of free space, \(\mu_0\), will be experimentally determined. \(\mu_0\) is a fundamental constant of nature, and, with its electric equivalent \(\varepsilon_0\) that we determined in an earlier experiment, we will now be able to determine the magnitude of the velocity of propagation of an electromagnetic wave — the speed of light. These constants are related to the speed of light, c, by the following relation, derived from Maxwell’s equations which describe the behavior of electric and magnetic fields:

(102)#\[c=\sqrt{\frac{1}{\varepsilon_0\mu_0}}\]

You will measure \(\mu_0\) here, and with your previous measurement of \(\varepsilon_0\), determine the speed of light \(c\).

The 2019 redefinition of SI units defined exact values of fundamental constants including the electron charge \(e\), the speed of light \(c\), and Planck’s constant \(h\). The second is defined in terms of the frequency of a Cesium atomic clock. As a result, the value of \(\mu_0\), the magnetic constant, must now be experimentally determined.

Note: Until 2019, \(\mu_0\) was defined to be exactly \(4\pi \times 10^{-7}\) T⋅m/A. This 2019 value is so close to the current best measurement, that you can use \(\mu_0=4\pi \times 10^{-7}\) T⋅m/A as the accepted value.

The magnetic field strength \(B\) a distance \(r\) from the center of a very long, straight conductor carrying a current \(I\) is given by:

(103)#\[B=\frac{\mu_0 I}{2\pi r}\]

A conductor of length \(L\) [m] carrying a current \(I\) [A] in a magnetic field of strength \(B\) [T] experiences a force \(F_B\) [N], given by:

(104)#\[\vec{F_B}=I \vec{L} \times \vec{B} =I L B \sin{\theta}\]

where \(\theta\) is the angle between the vectors \(\vec{B}\) and \(\vec{L}\). If the magnetic field is produced by the current in a second conductor, the currents in the two conductors exert equal magnitude and opposite forces on each other.

For the case of two long, parallel conductors, each carrying the same current \(I\) and separated, center-to-center, by the distance \(d\), the force between the two conductors is the force on one in the magnetic field of the second. Thus if (103) is used in (104), we obtain for the force:

(105)#\[F_B=\frac{\mu_0}{2\pi}\frac{L I^{2}}{d}\]

By measuring the force between two such conductors, the value of \(\mu_0\) can be determined. By considering the parameters of the apparatus to be used, it can be seen that the magnitude of the force between the two conductors is quite small and will require some care to accurately measure. If \(L\) is about 30 cm, \(I\) about 10 A and d about 2 mm the force is in the range of \(10^{-4}\) N, or the weight of a few milligrams of mass. Since this force is well below the weight of a reasonable conductor to be used for the experiment, a counter weighted balance system will be used to provide the mechanical support for the movable conductor. This is essentially the same apparatus as Experiment E-1. The movable conductor can then be loaded by a known mass whose weight can then be matched by the small magnetic force. The apparatus is schematically illustrated in Figure 57.

Apparatus dimensions, similar to the apparatus setup from E-1.

Fig. 57 Apparatus dimensions, similar to the apparatus setup from E-1.#

As before, similar to the setup and measurement methods from E-1, an optical lever, telescope and scale are used to observe the deflection of the balance, which carries the movable, upper conductor. The magnitude of the magnetic force can be measured when the magnetic force has restored the loaded system to its original position. In this case we make the same magnitude current flow in opposite directions in the two conductors so the force between the two conductors is repulsive. Thus on the upper conductor, there is a gravitational force down (the weight of the added mass) and a magnetic force upward which we will adjust by controlling the current equal to the gravitational force. More conveniently than E-1, this equilibrium is stable!

With the required balancing current determined along with the other dimensions of the apparatus, we can now determine the value of \(\mu_0\).

Equipment#

  • Telescope with crosshair & centimeter-scale ruler on vertical pole

  • AC power from wall outlet, controlled with large, cylindrical potentiometer. Controls range from 0 – 100% of wall power (~0 – 15 A). Ignore the scale on the dial since the knob is not actually lined up to them.

  • Voltage transformer to drop AC voltage to ~6 V (small metal cube)

  • Fluke multimeter – in series, set to read AC current (\(\tilde{\text{A}}\)), red wire to 10 A, black wire to COM ports

  • Small masses of 1 – 500 mg, plastic tweezers

  • Parallel-conductor apparatus

    • Bottom conductor held in static position

    • Top conductor that can swing up or down when changing amount of applied mass or current. Set up to be parallel to bottom conductor when empty (holding 0 mg).

    • Mirror to view the scale (ruler) via telescope to determine angle between plates

    • Beam lift knobs to reset top plate position

    • Leveling screws to level whole apparatus

  • x4 Banana plug wires (12 AWG, x2 1’, x2 3’) connecting wall power from transformer to ammeter & parallel-conductor apparatus

  • Protective box with the mirror-to-scale distance written on it

Adjustment of apparatus#

The apparatus has been carefully adjusted before your lab and should not require further significant adjustments. This section describes how the apparatus was prepared. If something seems to need adjusting, see the lab instructor. The apparatus is very similar to the apparatus setup from E-1.

  1. The beam lift provides a definite location for the beam and thus guarantees continued alignment of the parallel conductors. Use the beam lift each time you change weights or relocate the counterweight.

  2. The fixed conductor can be adjusted vertically, the movable conductor horizontally, so that the conductors are parallel. If the conductors are NOT parallel with no mass in place, seek the instructor’s help.

  3. The counterweight behind the mirror can be used to change the equilibrium separation of the conductors.

  4. Level the apparatus with the adjusting screws so that it sits securely on the table.

  5. Mirror-to-scale distance was measured from rear of mirror (reflective surface) to roughly the middle of \(S_1\) and \(S_0\) on the scale (ruler).

Experimental Procedure#

Danger

DO NOT TOUCH THE METAL CONDUCTORS (BARS)

PRECAUTIONS#

  1. The wire frame that supports the current carrying conductor and counterweight is supported on knife edges. The frame is easily bent, and the knife edges can be easily damaged. Treat the system with the same care as a precise analytical balance. Handle the weights with tweezers and store them in the case.

  2. The current must pass through the knife edges and intense local heating is produced. Reduce the current to zero as soon as possible after making the observations.

  3. Do not be the ground of the circuit.

  4. The current should not exceed 15 A!

Data Acquisition#

If \(b\) is the mirror-to-scale distance and \(a\) is the length of the frame (see Figure 57), then a straightforward geometrical analysis (like in Lab E-1) will show that the vertical displacement of the conductor, \(y\), is given by:

(106)#\[y=\frac{D a}{2 b}\]

where \(D\) is the change in scale reading from the equilibrium value and the value when the two conductors are in contact (the contact reading is determined by adding a mass to the pan which depresses the beam until contact occurs). The factor of 2 results from the fact that the optical path reflected off of the mirror to the scale rotates through an angle twice that of the beam holding the movable conductor.

However, the change in scale reading doesn’t account for the conductors’ thickness. Since the B-field generated by current flowing in the cylindrical conductors is organized about their center, we must add in two radii to get the center-center separation between the two conductors. This gives us:

(107)#\[d=\frac{D a}{2 b}+2r_\text{conductor}\]

where \(r_\text{conductor}=1.6\) mm is the radius of the conductor (multiplied by two to deal with the fact that we’re dealing with two conductors).

  1. Investigate the use of the telescope and scale so that the rotation of the frame can be measured in terms of scale divisions (see Figure 58).

Schematic of the measuring apparatus, similar to the setup and measurement methods from E-1.

Fig. 58 Schematic of the measuring apparatus, similar to the setup and measurement methods from E-1.#

  1. Do not turn on the power supply until requested in the procedure below.

  2. In order to measure the equilibrium separation distance, \(d\), of the conductors, two readings are made. The first is the scale reading, \(S_0\), at the equilibrium position, i.e. when no mass has been added to the pan on the movable conductor.

  3. The second scale reading, \(S_1\), is made when the top conductor contacts the lower conductor. Place a sufficient mass in the pan on the top conductor to make it contact the lower, stationary conductor. Record \(S_1\).

  4. Determine \(D = | S_1 - S_0 |\) from two scale readings (reminders: absolute value there is to represent the total distance between \(S_0\) and \(S_1\). If your crosshair crosses 0, make sure to consider the negative).

  5. Create a common data table with \(S_0\), \(S_1\), \(D\), length of frame \(a=0.215\) m, mirror-to-scale distance \(b\), wire length \(L = 0.265\) m, and the wire radius \(r_\text{conductor}=1.6\) mm.

  6. Referring to Figure 57 and Figure 58, use (107) to calculate the conductor separation distance, \(d\), then record it.

  7. Create a data table with columns for the applied mass, the applied force, the current \(I\), the square of the current \(I^2\), and \(\mu_0\) for each individual trial. You’ll have a row for average and standard deviation to use as well.

  8. With the power off, make sure the conductor beam is at equilibrium, then place the specified mass on the upper conductor.

  9. Make sure the transformer setting is zero before turning on the power. While observing the scale through the telescope, slowly increase the current until the scale returns to the equilibrium point \(S_0\). This is the current required to ‘balance’ the weight of the added mass. As soon as you make this reading, reduce the transformer setting to zero and turn off the power switch. Under no circumstances should the current exceed 15 A.

  10. DO NOT LEAVE THE POWER ON, IT CAN BURN OUT THE WIRES IF LEFT ON TOO LONG AND CAUSE INJURY

  11. Record your values for the current trial and calculate \(\mu_0\) from (105).

  12. With power off, replace the applied mass with the next trial, and repeat the previous steps to find the current required to balance the system again.

  13. After completing all 27 trials, calculate your experimental average and standard deviation (treated as the uncertainty range) of \(\mu_0\).

  14. If, after performing the graphical data analysis below, you find some or all of the data unacceptable, repeat from step 3, checking that the telescope and scale were not disturbed during your measurements.

Analysis of data#

  1. The graphical display of data permits the comparison of all the values and associated errors at once. Points that depart markedly from the general trend of the data are quickly detected. We expect from theory (from (105)) that:

    (108)#\[F_B=k I^{2}\]

    Using all your data points, scatter plot \(F_B\) as ordinate (\(y\)) and \(I^{2}\) as abscissa (\(x\)). Fit a straight line through the points and determine the slope of the curve, \(k\). Reordering (105), the value of \(k\) is given by:

    (109)#\[k=\frac{\mu_0}{2\pi}\frac{L }{d}\]

    \(\mu_0\) can now be determined by rearranging (109) and using this experimentally determined value \(k\). Thus:

    (110)#\[\mu_0=\frac{2\pi k d}{L}\]

  2. Using your experimentally-determined values of \(\varepsilon_0\) you determined in Experiment E-1 and the value of \(\mu_0\) determined above, calculate your estimate for the speed of light \(c\) with (102). You can estimate your uncertainty in \(c\) by varying \(\mu_0\) and \(\varepsilon_0\) by their own uncertainty ranges to see how much \(c\) varies.

Post-Lab Submission — Interpretation of Results#

  • Submit your finalized data sheet with summarized data and cleaned-up plots (Excel sheet)

  • Paragraph of your results ± uncertainties from your data. Make sure to include discussion of the following:

    • Compare your experimental values of \(\mu_0\) from plotting with (110) and the average ± std. dev. of your values from (105). Do they agree within your expected error?

    • Explain physically, why would you expect the linear fit to go through the origin?

    • Do your experimental results agree with the accepted values of \(\mu_0 = 1.2566\times 10^{-6}\) T⋅m/A?

    • Do your experimental results agree with the speed of light (\(c = 2.9979\times10^8\) m/s)?

  • Paragraph of your errors and estimated measurement uncertainties. Be quantitative. Make sure to include discussion of the following:

    • Where might systematic (affecting accuracy) and/or random (affecting precision) errors be coming from?

    • What are your measured uncertainties, and, based on these uncertainties, how do your final results change? I.e. do your different measurement and slope uncertainties make your final results larger or smaller?

    • Change different variables by your best estimation of measurement uncertainties in your Excel sheet; what variables affect the accuracy of your final results the most?

    • If larger or small, are they more or less accurate to expected values?

    • How could you improve your random errors?

    • Were your systematic errors significant; how could this be improved if you were to re-run this experiment?

  • Reflect on this week’s lab; what did you learn?