Mapping Electric and Potential Fields
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1 Mapping Electic and Potential Fields Pupose We detemined the magnitude and diection of the electic fields suounding thee sets of chaged electodes. Fo each electode set, we measued the electic potential suounding the electodes and plotted the lines of equal potential on gaph pape. Fom the data, we calculated the value of the electic field. Theoy The thee, two-dimensional electode sets ae diagamed below. The positive electode fo each was set at 15 volts and the negative electode was at gound potential. Ou pocedue was to find sets of points at the same potential suounding each electode. We did this at 3-volt intevals fom the negative to the positive electode in each set. Connecting points at the same potential with smooth cuves mapped out the potential field fo each set. The diection of the electic field fo each set was detemined by using the pinciple that the electic field lines ae pependicula to the equipotential lines, and un fom high potential to low. The thee esulting field maps ae in the data section below. To detemine the magnitude of the electic fields, we used the fact that the electic field is the negative ate of change of potential: E = -dv/d whee is the elevant positive coodinate. (This follows fom the definition of potential: V ( ) = E ds, which states that the potential at is the wok done against the electic field to place a unit chage thee.) Rectangula Electodes. Fo this set we found the electic field to be constant in diection in the cental egion between the electodes. Expecting the field to be constant in magnitude also, we used: ΔV E = Δx
2 to detemine the aveage value of the field in this egion. The position coodinate x was measued fom the negative electode. Concentic Electodes In this case we found the equipotentials to be oughly cicula, indicating a adially diected electic field. Since the distance between field lines inceases (and the magnitude of the field thus deceases) in popotion to the adial distance, the electic field fo such a two-dimensional configuation must have the fom V E = m whee V m is some constant voltage. In tun, this means the potential must have the fom: V ( ) = V a V m ln( ) a whee a is the adius of the inne electode and V a is the potential thee. (This can be dv Vm deived fom E = = and integating fom =a to abitay adius.) d We measued the aveage adii of the equipotential lines and plotted ln(/a) vesus the potential V. Fom the slope of this gaph, we detemined V m. Using this, we calculated the magnitude of the electic field at the adii of the equipotentials. Cicula Electodes. The potential lines hee wee ovoid, indicating a field that depends on both position coodinates: x and y (o and θ). We confined ouselves to calculating the field along the cental axis only using ΔV E( x) Δx whee the position coodinate x was measued fom the positive electode. We expect this to be an appoximation to the field values half way between each equipotential line.
3 Expeiment Set up The electodes consisted of two-dimensional egions dawn with metallic ink on conductive pape: Set 1 (ectangula) Two ectangles 12 cm x 1 cm, 9 cm apat Set 2 (concentic) Positive electode: a cicle of adius 1 cm Negative electode: a cicle of adius 8 cm Set 3 (cicles) Two cicles of adius 1 cm, centes 10 cm apat. A 15-volt potential diffeence was applied acoss the electodes with a low-voltage powe supply. We measued the potential using a VC 9808 Digital Multimete. The mete was connected to the negative electode and a voltage pobe was touched to the pape to test the voltage at the point of inteest. Fo each set, we found the 3-volt, 6-volt, 9-volt and 12-volt equipotential lines. Rectangula Concentic Cicula Data (See diagams below.)
4 Analysis The field maps can be seen below. The electic field calculations ae as follows. Rectangula electodes We measued the positions of the equipotentials along the cental line between the electodes. The position (x) was measued fom the positive electode. Fom this, we calculated aveage value of the field. The esults ae Potential (V) Position (cm) E-field (V/cm) deviation (V/cm) Aveage: (σ =.1) E-Field value: 1.7 ±. 1 volts/cm, o 170 ± 10 volts/mete Results have been ounded to 2 significant figues. Concentic Electodes The aveage values fo the equipotential adii ae listed in the table below. Also below is a semi-log plot of (/a) vesus the potential. The slope of the gaph, using the indicated values on the plot, is Slope = ln( 1.95/ 6.8) =.139volts 1 The y-intecept is ln(8). The equation of the gaph is ln(/a) = ln(8) V() Compaing this with V ( ) = V a Vm ln( ), we have a
5 V m = 1/.139 = 7.19 volts Using V(=8cm) = 0 volts, V a = 15 volts and = 8 cm, and plugging into the theoetical expession fo V(), we get V m = 7.21 volts (theoy) Ou empiical esults diffe fom theoy by 0.2%. The values of the electic field at the adii of the equipotentials wee calculated using ou empiical value fo V m. The esults ae below. V(volts) (cm) E (volts/cm) (ounded to 2 significant figues) Cicula Electodes We detemined the positions of the equipotentials on the cental line between the electodes. We appoximated the electic field stength at the midpoints between equipotentials by calculating the aveage value of the field between them (ΔV/Δx). x (cm) V(x)(volts) midpoint (cm) Electic Field (V/cm)
6 Conclusions The electic field between ectangula electodes is oughly constant in magnitude and diection in the egion away fom the ends of the electodes. The field bulges out at the end of the electodes. Fo vey long ectangles, we would expect the field to be vey unifom in the cental egion. We would expect this also to apply to ectangula plates o paallel sheets of any shape: unifom in the cente, with bulging at the edges. The concentic configuation led to adial fields, with the potential vaying logaithmically and the electic field going as 1/. The field was zeo outside the oute electode. The same should be tue fo cylindical electodes, as in coaxial cables. The cicula electodes led to ovoid potential lines. The field was stongest nea each electode, weake in the cental egion between them, and weakest fa fom the electodes (since the field lines become spase thee). We would expect any two localized egions of opposite chage, sepaated by a small distance, to have a simila field. (Such a field is called a dipole field.) Fo instance, a wate molecule is known to have this type of chage sepaation, and many of the popeties of wate ae due to the dipole field suounding its molecules.
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