Phys 2212L

LAB 2

Electric & Potential Fields Purpose An electric field surrounds any assemblage of charged objects. To determine the strength and direction of these fields, it is most convenient to first map the electric potential of the field, which can be measured with a voltmeter. From the potential field, the electric field can easily be determined. In this laboratory, we will map the electric and potential fields resulting from three different configurations of charged electrodes – rectangular, concentric, and circular.

Principles The electric field at a point in space shows the force that a unit charge would feel if it were placed at that point:

  F E= q E is a vector quantity: it has both magnitude and direction and has units of newtons per coulomb. The field is set up by electric charges somewhere in the surrounding space. For continuous charge distributions, it is much easier to analyze electric forces using the field concept than using Coulomb’s Law for the forces between point charges. If we measure the field (magnitude and direction) at enough points around the charge distribution, we could make a map of the electric field lines. These lines show the direction of the electric force at each point. However, instead of measuring the electric field directly, we will measure and map the electric potential around the charge distribution. The electric fields can be determined from this.

Electric Potential To place a positive test charge in an electric field we must do work against the field, since the field tries to push the charge away. Since the electric force is conservative, the work we do in placing the charge in the field is stored as potential energy. Electric potential is defined as the work done per unit charge to move the charge into an electric field. Like potential energy, electric potential is measured relative to some reference position, so that we define it as

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Electric & Potential Fields

(1)

∆V =

W q

where ∆V is the change in electric potential in going from the reference point to the point in question, W is the work done (or change in potential energy) and q is the charge. The unit of electric potential is the volt, which is equivalent to a joule per coulomb. We usually use the Earth itself as our reference point, and take the electric potential of the Earth to be zero. Equipotential Lines Thus a point in an electric field or in an electric circuit can be characterized by the electric potential, or simply the potential, there. A quantity of charge placed at that point has potential energy equal to the voltage there times the amount of charge. We can measure electric potential directly using a voltmeter. We can then map the potential field by connecting points that are at the same electric potential. Lines between such points are called equipotential lines. Moving a charge along an equipotential line requires no work, since the energy of the charge does not change. To find the direction of the electric field, we make use of the fact that the equipotential lines must always be perpendicular to the electric field lines. This is because the electric field lines show the direction of maximum decrease in the potential.

+

= E-field lines = Equipotential lines

Determining the magnitude of the Electric Field

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Electric & Potential Fields To determine the electric field from a series of voltage measurements, we note that the potential difference between two points is the integral of the electric field taken along the path between the two points:       W − ∫ F ⋅ ds − ∫ qE ⋅ ds ∆V = = = = − ∫ E ⋅ ds q q q where ds is an infinitesimal element of the path. The minus sign arises because the external agency must exert a force that is equal and opposite to the force exerted by the field, E. Conversely, the above expression tells us that E is the negative (vector) derivative of the potential – it gives the rate and direction of maximum decrease in V. In situations of symmetry, the vector notation simplifies. For instance, suppose E is constant in magnitude and direction, as between 2 oppositely charged electrodes:

+ E

Then the change in potential in moving a distance ∆x away from the positive plate is ∆V =

  W = − ∫ E ⋅ ds = − E∆x q

Conversely, the magnitude of the electric field can be determined from: (2)

E=−

dV ∆V =− dx ∆x

The minus sign signifies that E points in the direction that V decreases. Note also that the electric field can be expressed as volts per meter. Now consider a radial electric field in two dimensions, as from a disk of charge of radius a:

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Electric & Potential Fields

Here, the E-field is the same in all radial directions in the plane. Its magnitude must decrease inversely with the radius r from the center of the disk, since the field lines become sparser in proportion to r. The magnitude of the field can then be expressed as E=

(3)

Vm r

where Vm is a constant. The direction of the field is radially outward. The change in potential in going from radius a to some arbitrary radius r is then (4)

  V r ∆V = − ∫ E ⋅ ds = − ∫ m dr = −Vm ln( ) r a

Conversely, if we know the potential has this form, we can get E by taking the derivative of (4): (5)

E=−

dV Vm = dr r

In more general cases, where the field is not symmetric, or when the symmetry involves more than one coordinate, the derivative must be taken with respect to each spatial coordinate. However, over small distances, one can treat the field as approximately linear and can approximate: (6)

E ( x) ≈ −

∆V ∆x

where x is the relevant coordinate.

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Electric & Potential Fields

Electrodes Sheets Rectangular 1 cm 9 cm 12 cm

Circular 1 cm radius

10 cm center to center

Concentric

8 cm radius

1 cm radius

a = radius of inner electrode = 1 cm b = inner radius of outer electrode = 8 cm

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Electric & Potential Fields

LAB 1

Procedures We will measure the potential and electric fields around the three sets of electrodes illustrated. The electrodes are drawn on sheets of conductive paper with conductive ink. The procedure will be as follows: • • •

Measure the potential at representative points on the electrode sheets. (We often speak loosely of measuring the “voltage” rather than the potential.). Plot the values on graph paper. Draw the equipotential lines. That is, draw the best smooth curve through points at the same potential. The result will be something like a topographic map for the electric field. Draw the electric field lines by starting at the positive (high potential) electrode and tracing a path to the negative electrode in such a way that the electric field lines always cross the potential field lines at right angles.

Equipment • • • •

Low Voltage Power Supply (LVPS) Digital Multimeter (DMM) Conductive paper electrode sets: rectangular, circular and concentric Metal pushpins (2)

• • • • •

Corkboard Banana wires Alligator clips Plastic pushpins Graph paper

1. Set-up •

Make electrode sheets. If electrode sheets are not already made up, you can make them by drawing the electrodes on conductive paper using a conductive ink pen. Simply outline the electrodes in the proper dimensions and fill in the outlines with the conductive ink. Take care to make the facing edges of the two electrodes as smooth as possible – use rulers or circle templates. The exterior edges need not be smooth.

•

Set up a diagram on graph paper. Draw a horizontal and vertical axis through the center of a sheet of centimeter graph paper. Draw the outlines of the electrodes on the graph paper at a 1:1 scale. Place the origin of the diagram in the center of the electrode arrangement.

•

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Connect the electrodes to the power source.

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Procedures

1. Tack or pin the electrode sheet to a corkboard or other backing with plastic pushpins. Use the four corners of the conductive paper. 2. Insert a metallic pushpin or other conductor into the center of the positive and negative electrodes. For the outer circle in the concentric set, the pin can go anywhere within the outer circle. 3. With the power supply off, connect one electrode to the ground terminal of the DC output on the power supply with an alligator clip and banana wire or other connector. Connect the other electrode to the positive terminal on the power supply. 4. Set the current knob on the power supply to one half turn and set its voltage knob to the lowest setting (counterclockwise). Have your instructor inspect your set-up before you turn on the power supply.

Review the section on electrical measurements in the introduction before proceeding. Make sure you know the proper settings for the DMM and how to use it. 5. Turn on the power supply and set the voltage output to 15 VDC. Use the DMM connected across the power supply’s terminal to set the source voltage, not the meter on the power supply. 6. Take voltage readings as directed below. Connect the “COM” terminal on the DMM to ground (the negative terminal) on the power supply. Touch the DMM voltage probe to a point on the conductive sheet to measure the voltage there. It is not necessary to puncture the sheet. Take the reading to the nearest 0.1 volt and mark the corresponding point on your graph paper diagram. Label the point with the voltage reading.

2. Rectangular electrodes 1. Starting at the negative electrode (this should be at 0.0 volts) find the equipotential lines at every 3 volts. That is, find the 3.0 V, 6.0 V, 9.0 V, and 12.0 V equipotential lines. The edge of the positive electrode should be at 15.0 V. To find a line, slowly drag the probe away from the negative electrode until the DMM displays the voltage you seek. Keep the probe vertical, and allow the DMM enough time to settle down before you take the reading. Mark and label the point on your graph paper diagram. Repeat this process for about 8 evenly spaced points at the same potential, including the region beyond the ends of the electrodes. You should be able to trace the equipotential from one edge of the paper to the other.

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Procedures

Analysis 1. Draw the best smooth curves through the points at the same potential. These are the equipotential lines. 2. Draw the electric field lines. Starting at the positive electrode on your diagram, draw a line to the negative electrode in such a way that the line always crosses an equipotential line at right angles. Draw 6-9 field lines through the region between the electrodes and the regions above and below their ends. Use arrows on the lines to show the direction of the electric field. 3. Determine the magnitude of the electric field strength in the central region between the electrodes by calculating E=

∆V ∆x

for each adjacent pair of equipotential lines. Let x be the distance from the positive electrode. Find the average of the values – this is the average value of E in this central region. 4. Calculate the deviation for each value and find the average deviation. 5. Calculate the standard deviation and report the value of the field in the form: E = E average ± σ

3. Concentric electrodes Connect the inner ring electrode to the positive terminal on the power supply and the outer ring to ground. 1. Find the 3.0 V, 6.0 V, 9.0 V and 12.0 V equipotential lines and plot them on your graph paper diagram. Find 8 ts for each line in the region between the electrodes. 2. The field should be zero in the region outside the larger circle. Test this by finding the potential at 4 symmetric points outside the outer electrode. Analysis 1. Draw the best smooth curve through the points at the same potential in the interior region. 2. Draw 8 electric field lines in the interior region. 3. Determine the average radius for each equipotential line and tabulate the values of the potential with the average radii. Measure the radius from the center of the electrode

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Procedures

4. 5. 6.

7.

sheet, so that the outer radius of the positive electrode is at r = 1 cm and the inner radius of the ground electrode is at r = 8 cm. Plot r/a versus V on semi-log graph paper. (Refer to equation 4 above. In our case, a = 1 cm.) See the section on semi-log plots in the appendix. Determine the slope and the y-intercept and write down the equation of the line. Solve the equation for V(r). (That is, rearrange the equation so that V is on the left and all other terms on the right.) Evaluate the numerical terms. You now have an empirical equation for V as a function of r. Equation (4) above can be written r V (r ) =V a−Vm ln( ) a

(4)

Comparing your empirical equation with this, determine the value of Vm. Also, find Vm directly from this expression using Vb = 0 and compare with your empirical result. 8. Using your value of Vm, calculate the electric field strength at the radii of each of the equipotential lines.

3. Circular electrodes Find the 3.0 V, 6.0 V, 9.0 V and 12.0 V equipotential lines and plot them on the diagram. Your plot should fill the diagram – take readings both between and on the far side of the electrodes. Analysis 1. Draw the best smooth curve through the points at the same potential. 2. Draw 8 electric field lines, starting at 8 symmetric points on the positive electrode and ending at the negative electrode. 3. Along the central axis between the electrodes, the field should be a straight line, although it is not constant in magnitude. We can find an approximate value ∆V for the field at points along this central axis by using: magnitude E ≈ for ∆x closely spaced points. Use this to find the approximate field strength between each adjacent pair of equipotential lines. (Don’t forget that the surfaces of the electrodes are also equipotentials.) 1. Where is the field strongest and where weakest along this central line? 2. Is the field stronger near the positive or near the negative electrode?

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