Introductory Laboratory, Faculty of Physics and Geosciences, University of Leipzig

E 5e Electric Fields

Tasks

1 The voltage distribution between two concentric electrodes (model of cylinder capacitor) has to be measured! The measured radial electric field shall be compared with theory and the field distribution shall be discussed.

2 Determine the electric equipotential lines for a model of a lightning conductor! The discussion of the electric field lines shall be carried out by means of graphic representation.

3 The dependence of the voltage along a straight connecting line between two disc shaped electrodes shall be measured and compared with two theoretical models!

Literature

Physics, P. A. Tipler, 3rd Edition, Vol. 2, Chapt. 18-4, 18-5, 19-2, 19-3, 20-1... 20-5 Physikalisches Praktikum, Hrsg. D. Geschke, 12th Edition (in German), , Chapt. 2.0.1, 2.1 http://theory.uwinnipeg.ca/physics/charge/ http://hyperphysics.phy-astr.gsu.edu/hbase/electric/elefie.html

Accessories laboratory power supply, digital multimeter, different electrode models

Keywords for preparation

- definition of electric field, electrical flux density, displacement density, Gauss' law - electric potential concept of electrostatics, relationship between electric field and potential - image representation of equipotential surfaces and field lines - influence of dielectrics on electric field lines, dielectric breakdown

Hints to basics

Various vector fields, like electrostatic fields, current density fields in homogeneous conductors but also fields of flow in incompressible liquids, are similar to each other, if there are no sources. Therefore they can be described by the same differential equations. For electrostatic fields (no charges in the field, no vorteces) it is valid for the field strength E: divE = 0, rot E = 0 (∫Eds = 0). From this follows E = -grad ϕ (ϕ scalar potential) and ∇2ϕ = 0 (Laplace`s ). Completely analogous equations can be formulated for the current density j in homogeneous conductors using the constant conductivity σ in case of absence of vorteces, i.e. absence of 2 variable magnetic fields: j = σE, div j = 0, rot j = 0. From that follows j = -gradϕj , ∇ ϕj = 0. In many practical cases there are no closed mathematical (numeric) solutions. Often model experiments help in such cases in which the potential distribution (current density distribution) of plane (two- dimensional) electrode configurations are determined whereat then the corresponding three-dimensional fields must have sufficient symmetries. In practice one often uses conducting special paper as two- dimensional conductor on which metal electrodes are screwed or painted by a silver solution. In this experiment the potential distributions (field distributions) shall be measured for different electrode configurations. The electrodes are connected to a constant predefined voltage 10V. In order to determine the potential course the surface of the contact paper is scanned by a sensor and the voltage difference

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between one of the both electrodes and a point between them is measured by a high-impedance voltmeter (internal resistance Ri >10 MΩ). The voltage value corresponds to the electric potential at the test point with respect to the electrode connected with the measuring instrument. By this way one gets the equipotential lines which actually represent surfaces of the same potential. Then the electric field lines are the set of curves oriented perpendicular to these surfaces. In the different experimental setups the electrodes shall be regarded as equipotential surfaces which have the same electric potential. Practically this condition is fulfilled because the electrical conductivity between the electrodes is negligible in comparison to the conductivity of the electrode materials. In order to avoid damaging of the contact paper this shall not be touched by fingers and shall not be scratched during scanning process! For the most simple electrode configuration consisting of two parallel plane electrodes the potential distribution in the area of the homogeneous field can be described by a linear equation of the general form

Ux()= ∆=+ϕ () x a b x . (1)

At task 1 the voltage distribution (potential distribution) has to be measured in the model of cylinder capacitor with respect to the inside electrode. The model consists of two concentric electrodes, the inner circular disc with the radius ri and the outer circular ring with the radius ra. The voltage U (10 V) is applied to the electrodes. It has to be expected, that the equipotential lines are concentric circles around the center of the inner electrode. Applying the potential theory to derive the voltage dependence U (r) it follows

U 0 ⎛ r ⎞ U (r) = ln⎜ ⎟ . (2) ⎛ ra ⎞ ⎝ ri ⎠ ln⎜ ⎟ ⎝ ri ⎠

Fig. 1 (schematic) To the measurement of the equipotential surfaces for the model of a cylinder capacitor

Derive the equ.(2) using Gauss' law! One gets the radial distribution of the electric field in the cylinder capacitor according to equ.(2) from the derivation of dU(r)/dr. Due to the radial symmetry of the arrangement it is enough to carry out the measurements along to an arbitrary radius. In order to increase the accuracy of the measurements shall be carried out along to four different radii beginning at the center to the outside. The averaged voltage values shall be represented graphically in a single-logarithmic coordinate system. Using the voltage U0 and the radii ri and ra the theoretical electric potential shall be calculated according to equ.(2) and shall be represented in the diagram (graphic representation and comparison with calculable curve or data fitting by means of ORIGIN).

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At task 2 the electric equipotential lines between a point-shaped electrode placed oppositely to a plane electrode has to be determined (two-dimensional model of lightning-conductor Fig. 2) for U=10V. At appropriate points the potential lines have to be measured in little distances, in order to indicate the characteristical equipotential (electric field) lines close the edge. Along to the symmetry axis s between the point electrode and the plane electrode further values U (s) have to be measured. The equipotential lines as well as the field lines have to be represented graphically. The change of the electric field near the point-shaped electrode shall be discussed creating a diagram voltage U versus distance s (eventually using the slope dU(s)/ds calculated by ORIGIN, differentiation).

Fig. 2 Model of lightning conductor

At task 3 the voltage U (x) has to be measured along the straight connecting line (coordinate x) between two oppositely charged disc shaped electrodes. It has to be checked, whether the measured voltages correspond to equ.(3) derived that the equipotential surfaces due to a spherical surface of the electrodes 1 ⎡ ⎛ 1 1 ⎞ r(d − r) ⎤ U (x) = U 0 ⎢ ⎜ − ⎟ + 1⎥ (3) 2 ⎣ ⎝ x d − x ⎠ 2 r − d ⎦ or the equipotential surfaces can be explained by disc shaped (two-dimensional) electrodes

⎡ ⎛ d − x ⎞ ⎤ ⎢ln⎜ ⎟ ⎥ 1 ⎝ x ⎠ U (x) = U 0 ⎢ + 1⎥ . (4) 2 ⎢ ⎛ r ⎞ ⎥ ⎢ln⎜ ⎟ ⎥ ⎣ ⎝ d − r ⎠ ⎦

The radius of the electrodes is r and d is the distance between the center of the electrodes. The measured values U (x) have to be represented graphically and the both according to eqs. (3) and (4) calculable values have to be included as curves into the diagram.

Fig. 3 Computer 3d-plot of the electrostatic potential in the plane of an electric dipole. The potential due to each charge is proportional to the charge and inversely proportional to the distance from the charge.

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Discuss the different field line distributions

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