REPORT FOR LAB E3: ELECTRIC FIELD PLOTTING

Nathaniel Kan1/30/02

ABSTRACT:

In the Electric Field Plotting lab we used charged electrodes to create an electric field on a flat sheet of conducting paper. We tested several simple electrode positions and drew equipotential lines based on the voltage difference. We then confirmed that the equipotential lines run perpendicular to electric field lines by looking at the experimental data points.

PURPOSE:

The purpose of this lab was to familiarize ourselves with electric field plotting, by measuring the potential distributions and gradients for several different arrangements of electrodes on a sheet of conducting paper. We then looked for a relationship between the equipotential lines and the electric field lines. We then use LaPlace’s equation to calculate the potential distributions for some simple electric fields.

PRINCIPLES:

Electric charges create a space around them defined as an Electric field, which apply a force on other electric charges. This force is proportional and in the same direction as the electric field, and is defined as:

Fq = q E

Where q is a positive test charge which is small enough not to significantly affect the electric field. By moving a differential amount along the direction of the electric field at a point, then recalculating the new field vector, we can draw electric field lines, which plot the course of the electric field.

When moving a charge q a distance dr in an electric field there is a change in potential energy dV. This change is defined as equal to the work done divided by the charge q.

dW = -F · dr = -Eq · dr = -|E| q |dr| cos (Theta)

When dr is at a right angle to E, there is no work done and the potential, V, remains constant. By moving along a path on which is perpendicular to the electric field lines, we can draw equipotential surfaces where V remains constant.

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PROCEDURE:

We began by setting a basic “uniform” electric field. Laying down a sheet of uniform conducting paper with a resistance of around 5 – 20 kOhms, we placed two bar shaped electrodes down parallel to each other (see lab book page 52), bolting them down securely to prevent motion during measurements. The electrodes were then connected to a power supply to create a positive and a negative source of charge. The power supply was turned to its maximum capacity of 19.3V.

A voltmeter was used to test the potentials at different points in the 2D plane. The voltmeter had an input resistance of roughly 10 MOhms, resulting in field distortions of less than 1%. Procedurally, the voltmeter probe was used to find points roughly equal distance from each other with the same potential, and then a mark was made at each of the points. After drawing across the entire conductive sheet, points with the same potential were connected to form equipotential lines.

For the two parallel bars “uniform field,” we started with looking for the 10V equipotential, which would have cut the conductive paper approximately in half, according to symmetry. After that equipotential was drawn we measured points at other equipotentials, attempting to cover a range from around 0V to 20V.

This process of drawing equipotentials was repeated for five more electrical field geometries, including a dipole field, coaxial cylinders, a quadrupole field, a potential mirror, and a quadrupole mirror. Diagrams of these geometries are on page 52 of the lab book.

CALCULATIONS:

By dividing the medium of the electric field into a square grid with uniform spacing (∆x = ∆y), LaPlace’s equation can be solved to show that:

V(x,y) = (1/4)[V(x+∆x,y) + V(x-∆x,y) + V(x,y+∆y) + V(x,y-∆y)](See Lab packet E3 pg. 5)

We used a computer spreadsheet (LaPlace 1 and LaPlace 2) to run a simulation of the two parallel bar uniform field electric field. We know the potentials at the electrodes, and by applying LaPlace’s approximation and iterating many times, eventually the solutions for V will converge to something near the actually potentials. The accuracy increases as the size of the individual grid spaces (∆x and ∆y) decrease.

RESULTS:

We took data for all six electrode geometries. We found that the equipotentials are related to the electrode shapes and formations. For example, in the uniform field geometry, where there are two parallel bars, the equipotentials are straight lines parallel to the bars across the space between the two electrodes. They slightly curve around the ends of the

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electrodes as measurements are taken closer to the ends. This is a result of the bars being of a finite length. For a truly uniform field, the lines of charge would have to be infinitely long.

For all formations, we observed definite symmetry of the equipotentials. For the dipole field, which consisted of two “point” charges (one negative and one positive) spaced apart from each other, the equipotentials were rings around each charge, and as measurements were taken closer to the center, the rings became more oval and eventually became a straight line midway between the two charges. This confirms that a test charge very close to one of the electrodes will behave as if it is only under the influence of that point charge, but as the test charge moves closer to the midpoint between the two, it behaves as if it is between a dipole.

All geometries yielded fairly predictable shapes. The cylindrical one, which consisted of a point charge in the middle of a ring of charge yielded roughly circular equipotentials which radiated out from the point charge into the ring. This just confirmed that the electric field lines would radiate from the center positive test charge straight out into the negative ring. The quadrupole field created equipotentials which were hyperbolic, along the diagonals. The quadrupole mirror create equipotentials which were hyperbolic along one of the half circles, without the use of the other three half circles of charge. The potential mirror just a grounded conductor at the midpoint of a dipole. This created equipotentials roughly equal to those of the dipole, as expected.

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