E 7E “Magnetic Fields in Coils”

Fakultät für Physik und Geowissenschaften Physikalisches Grundpraktikum

E 7e “Magnetic Fields in Coils”

Tasks

1. Check the calibration of the teslameter.

2. Measure the field distribution along the coil axis of a solenoid. Plot the measured and calculated field distribution. Determine the magnetic induction at the end of the solenoid and compare to the calculated value.

3. Measure the axial field distribution of a circular coil. Determine graphically and arithmetically the distance from the coil center, at which the maximum value of the magnetic flux density is decreased to one half.

4. Measure the axial field distributions of a pair of circular coils for three different distances between the coils. Plot experimental and calculated field distributions in one diagram. Discuss discrepancies.

Additional task: Determine the magnetic dipole moment of a permanent magnet by rotational oscillations in a uniform magnetic field generated by a pair of Helmholtz coils.

Literature

Physics, P. A. Tipler, 3rd Edition, Vol. 2, 24-1, 25-2, 25-4 Physikalisches Praktikum, 13. Auflage, Hrsg. W. Schenk, F. Kremer, Elektrizitätslehre, 2.0.2, 2.2

Accessories laboratory power supply, digital multimeter, teslameter with axial field sensor (Hall sensor), various coils, ruler

Keywords for preparation

- definition of magnetic field strength and of magnetic flux density

- Amperes law, Biot-Savarts law, Maxwells equations for stationary fields - magnetic fields inside of coils (cylindrical coil (solenoid), circular coil, pair of coils) - magnetic pole strength and magnetic dipole moment, magnetic dipoles in a magnetic field - basic principle of Hall-sensor

Parameters of the various coils (relative uncertainty in the geometrical constants L and R: 1,5 %)

Coil Coil parameters

Calibration coil (Kalibrierspule) Length L = 161,4 mm (Solenoid) Winding number N = 300

Imax = 1.2 A Radius R = 16.3 mm Solenoid 1 Length L = 348 mm

Imax = 2 A Winding number N = 291 Radius R = 13.1 mm Solenoid 2 Length L = 300 mm

Imax = 2 A Winding number N = 282 Radius R = 20.4 mm Solenoid 3 Length L = 160 mm

Imax = 4 A Winding number N = 150 Radius R = 13 mm Solenoid 4 Length L = 300 mm

Imax = 1 A Winding number N = 625 Radius R = 9.2 mm Pair of coils 1(1) Winding number N = 390

Imax = 2 A Radius R = 9.6 cm Pair of coils 2(1) Winding number N = 102

Imax = 3 A Radius R = 15.35 cm Pair of coils 3(1) Winding number N = 320

Imax = 3 A Radius R = 6.8 cm (1) two identical flat circular coils

Basic principles

(Vectors are printed using bold letters in the text.)

1. Calculation of the field of a cylindrical coil (Solenoid)

The integral form of the Maxwell equation is      d   Hd s  j  d A   DAd  , (1)   A S dt   where S denotes any closed curve, A the corresponding enclosed area, H the magnetic field, j the current density and D the dielectric displacement. For DC currents one has

   Hd s   j  d A , (2) S 2

with the differential form   H  j . (3)

With the quantities defined in Fig. 1 Biot-Savarts law can be written as    I dl r dH  , 4 r2 (4)

Fig.1 Towards the law of Biot-Savart

 where r denotes the unit vector pointing from the line element dl to the observation point P; r is the distance between line element and P. Using this law the magnetic field strength, defined by the electrical current, can be calculated at any point P. Biot-Savarts law is the equivalent to Amperes law. In the special case of a circular conducting wire loop (see Fig. 2) the axial components (dHz) of all wire loop elements dl have the same direction and add up. However, the radial components dHr compensate in pairs.

Fig. 2 Calculation of the axial component of the magnetic field of a circular conducting wire loop using Biot-Savarts law

With dH = dHz /sin and sin =R/r one obtains

I Rd l dHz  , (5)  2 2 3 4 R z  since r is perpendicular to dl. The integration over all elements of the circular wire loop (full length 2R) yields the equation

IR2 H() z  . (6) 3 2 R2 z 2 

If the wire loops are adjacent to form a coil, the magnetic field components are superimposed to yield a total field at the test point P of distance z = a in Fig. 3.

Fig. 3

Calculation of the B-field of a solenoid

The density of turns (number of turns per unit length) of a long, tightly wound solenoid is n = N/L , where N is the number of turns and L is the length of the solenoid. Thus, ndz wire loops are between z and z+dz carrying the current I n dz. This current generates an axial component of the magnetic flux density dBz in the point P ( z = a) of Fig. 3

0 2 1 dBz  I R n d z , (7) 2 2 2 3 R() a  z  where 0 denotes the vacuum permeability.

The integration over the total length L of the solenoid gives

 a L a  B B() a 0 I n    . (8) z 2 2 2 2 2 R a R () L  a 

It follows from Eq. (8) that the amount of magnetic flux density at the beginning (a = 0) and at the end a = L of the solenoid is half of that in the center (a = L/2) of the solenoid (for R << L).

2. Calculation of the axial field component of a pair of coils

The starting point is the calculation of the axial component B(z) of the magnetic flux density of a flat circular coil with radius R and the number of turns N. Using Eq. (6) B(z) can be calculated immediately by adding the field contributions of the single wire loops for a flat coil with thickness D<< R):

 INR2 1 B() z  0 . (9) 2 ()z2 R 2 3

Fig.4 Calculation of B(z) for a pair of coils

The magnetic flux density along the axis of two identical flat coils facing each other parallel and coaxial, see Fig. 4, is calculated adding the fields of the single coils. For the coil distance b the magnetic flux density in distance z from the axis center (Fig. 4) is

    2    INR 1 1  B() z 0  . (10) 3 3  2 2 2  2b  2  b  R z   R   z    2   2  

The current direction is the same in both coils. Discuss the field distribution in case of oppositely directed currents. Check your consideration in the experiment. Using Eq. (10) one can calculate, that the magnetic flux density Bz shows a maximum for z = 0 and a  R , whereas for a  R it shows a minimum at z = 0. For a = R the magnetic flux density between the coils is approximately homogeneously in the range -R/2  z  R/2.

Hints to the experimental procedure

Task 1: Before starting the measurements, check the calibration of the digital teslameter. To this end the field sensor is positioned in the center of the solenoid provided for the calibration (data: number of turns N = 300, length L = 161.4 mm, diameter D = 32.6 mm). During the preparation at home, calculate the current necessary to generate a magnetic flux density of Bset = 2 mT at the center of the solenoid.

The teslameter needs about 10 min to warm up. After warm-up the zero point controller of the teslameter should be adjusted to zero. Then set the dc current value (measured by a digital multimeter, error 0,1 %) calculated during the preparation and read off the measured value of Bexp.

The relative standard error of the digital teslameter is 2 %. The difference between Bexp and Bset should not exceed a relative deviation of 3 %, otherwise a systematic correction should be considered.

Task 2: A coil current corresponding to a value B = 1 mT (or 2 mT) at the center of the solenoid has to be adjusted dependent on the maximal value of the current given at the work station. To measure the axial field distribution Bz(a) the distance a between the field sensor and the coil center is increased shifting the coil in steps of a magnitude appropriate to map out the field change with sufficient accuracy. Since the axial field distribution is symmetric around the center of the solenoid, the field distribution has to be measured only in one direction. Measured and calculated values for the field distribution should be presented in one diagram Bz(a).

Task 3: The axial field distribution of a flat circular coil (data given at workstation) should be recorded step by step analogous to task 2 but on both sides. In order to quantify the analysis plot both experimental and calculated field profiles in one figure. Determine graphically and arithmetically the distance from the coil center, at which the maximum value of the magnetic flux density decreases to one half of the value at the center of the coil.

Task 4: The axial field profiles of the magnetic flux density B(z) in a pair of coils have to be recorded step by step analogously to task 3. Start with the smallest distance a=R/2 and set the center value of the magnetic flux density to Bcent=2 mT (or higher dependent on the maximal current of the coils used). Additionally, measure the axial profiles B(z) for coil distances a=R and a=2R between the two flat coils (coil data as in task 3).

Additional task: magnetic moment of a permanent magnet

The magnetic moment m (units of A m2) is defined by the following equation:   M m  B (11) where M denotes the torque and B the magnetic induction.

A permanent magnet (magnetic moment m) suspended on a thin thread is brought into a homogeneous magnetic field BSp, which in our case is generated by a Helmholtz pair of coils. Other external magnetic fields, e.g. the horizontal component of the Earth’s magnetic field, can be neglected. After deflecting the magnet away from the magnetic field axis of the pair of coils it performs oscillations with angle . Neglecting friction forces as well as the restoring torque of the thread the equation of motion is   J  m BSp sin   0 ,   (,)m BSp . (12)

In Eq. (11) J denotes the moment of inertia of the permanent magnet. One obtains for the period

J T  2 . (13) mBSp

For the determination of the magnetic moment it is favorable to plot the dependence of the period -2 on the magnetic induction in the form T = f (BSp ) and to calculate the magnetic moment from the 6

slope of the curve. To this end the moment of inertia of the permanent magnet has to be known. The magnetic induction is calculated from the axial component of the magnetic field at the center of the

Helmholtz pair of coils ( a = R ) with BSp= 0.71550 N I / R, where the current is to be measured.

After fixing the coil distance to a=R and energizing the coils, the magnetic moment is suspended from a thread, placed in the center of the pair of coils and excited to rotary oscillations. Measure the -2 period for about five appropriately chosen magnetic inductions. From a plot T =f (BSp) determine the slope and calculate the magnetic moment in units of Am2.

Disk magnet Rod magnet Mass m = (14.01  0.02) g Mass m = (12.12  0.01) g Radius R = (12.6  0.1) mm Radius R = (2.95  0.01) mm Height h = (6.30  0.01) mm Length l = (60.1  0.1) mm

The moment of inertia of a homogeneous cylinder (mass m, length l, radius R) around its rotation axis 1 is J mR2 and around its center-of-mass axis perpendicular to the rotation axis is 2 R2 l 2  Js  m  . 4 12 