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FYSP105 / K2 HELMHOLTZ COILS Introduction In the exercise you’ll measure the spatial distribution of magnetic field of Helmholtz coils at various currents applied to them. Electromagnetics in general as well as Helmholtz’s coils can be found in: • Grant, I. S. & Phillips, W. R. Electromagnetism. 2nd edition, John Wiley & Sons, West Sussex, UK, 1990. Ch. 4 (especially pages 139-144) • Young, H. & Freedman, R,. University Physics with Modern Physics, 11th ed., Addison Wesley Longman, 2000. Ch. 28.5 and example 29.10. 1. Theory 1.1. Magnetic field of a wire Ampère’s law defines the relation between the integral of a magnetic field B around a closed path C and the current intercepted by the area. A spanning the path C: r r r r ∫ B ⋅ ds = µ ∫ j ⋅ da C 0 A (1) r Here j is the density of the electric current and µ 0 = 1,26 *10 −6 Vs Am is the permeability constant. Thus there is a relation between the total current I r r I = ∫ j ⋅ da (2) A r and the magnetic field B caused by it. The relation FYSP105 / K2 - 2 - → → ∫ B⋅ da = 0 A' (3) → (Gauss’ Law) on the other hand indicates that the flux lines of field B are continuous when A’ is a closed surface. Near a wire, the flux lines act as in Fig. 1, they have no beginning or end. → Figure 1: Magnetic field B , close to a wire with current I. Biot-Savart’s law applied to a finitely long piece of an electric wire gives the formula (4) → for the magnetic field at a distance r (compare with Fig. 2).. r r r r µ 0 Idl × (r − r ') dB = r r3 (4) 4π r − r ' r r Figure 2: Magnetic field dB at point P caused by a piece dl at point Q of an electric wire. FYSP105 / K2 - 3 - 1.2. Magnetic field of an infinitely long electric wire The magnitude of the magnetic field B at a distance r from an infinitely long electric wire can be derived from eq. (4). The result is µ0 I B= . (5) 2π r 1.3. Magnetic field of electric wire loop Biot-Savart’s law applied to an electric wire loop with radius R and current I, gives the magnetic field at distance x at point Q (Fig. 3) µ 0 IR 2 B= . (6) 2( R 2 + x 2 ) 3 / 2 If the loop (a coil or a solenoid) has N turns, the magnetic field is the sum of fields of individual loops i.e. the total field is N times eq. (6). Figure 3: Electric wire loop with radius R 1.4. Magnetic field of axial coils The total magnetic field of two identical axial electric coils (separation L, fig. (4), notice the selection of positive x-direction) at a distance x can be derived from eq (6): µ 0 NIR 2 1 1 B= ( R + x 2 ) 3 / 2 + ( R 2 + ( L − x) 2 ) 3 / 2 2 . (7) 2 FYSP105 / K2 - 4 - Eq. (7) is valid between the coils as well. Point Q must lie on the axis. Figure 4. Two identical, axial coils with separation L Problem: Two axial, identical coils with radius R at a distance L=R form a s.c. Helmholtz coils. In the experiment you'll use such a pair. Work out the problem a) before coming for measurements. To solve problem b) is not mandatory but to ponder about it is very useful. Can you observe any analogia? a) Sketch the magnitude of the magnetic field B(x), on the axis of Helmholtz coils between x = [–2L, 3L]. The origin is selected as shown in fig. 4. Use units µ 0 NIR 2 [B] = . 2 b) How does B behave at a non-axial point Q? 2. Equipment The experiment will be done with coils at distance L = 150 mm having diameter 155 mm, turns N =125 each and with equal currents I. The direct current can be obtained e.g. from Metric LPS 303. Measure B with e.g. Phywe Teslameter. Notice that the active part of the probe is its tip, functioning properly only in the direction of its axis. In other directions with respect to direction of B you'll observe the component of B in the corresponding angle. FYSP105 / K2 - 5 - 3. Measurements First set the reading of the magnetic field probe to zero using the fine tuning button on the right hand side of Phywe Teslameter. If this is not enough, use the small button on lower left corner. Use settings for uniform field and the most sensitive scale (20 mT). Then switch on the current for coils. Notice that the coils may heat up and thus the current may not stay constant. Keep eye on the current to avoid systematic errors. Perform the following measurements: • Magnetic field B on the axis, in the middle with several currents I ≤ 3 A. • Magnetic field B in several positions Q on the axis, between the coils with a constant current e.g. I = 3 A. Outside the coils you need to measure B only in one axial side (why?). • Magnetic field B in several non-axial points Q around the middle of the coils and in 1 - 2 interesting non-axial points Q. You may choose points 1 - 2 outside the coils as well. 4. Analysis of results Verify how well eq. (7) is valid for different values of x and I keeping one of them constant and varying the other. Compare measured values to theoretical ones. Draw the theoretical flux densities B(I) (x = constant) and B(x) (I = constant). To the same co-ordinates plot also the measured values. Do not forget the estimates for calculated and experimental errors. How does the non-axial field behave? Try to explain the phenomenon theoretically. Can you figure out any applications for this?