Purpose

Part A. To gain experience with equipotential lines and electric fields by experimentally determining the potential at various points between test electrodes and drawing the equipotential lines and the electric field lines.

Part B. To study the superposition of magnetic fields by mapping the field of a bar magnet combined with the earth's magnetic field.

Apparatus

Part A. Electric potential mapping board, conducting paper with painted electrode configurations, graph paper, -500 to 0 to 500 microampere meter, 0-5 V DC voltmeter, switch, power supply.

Part B. Bar magnet, two magnetic compasses, ruler, two large sheets of paper, tape.

You must bring your own graph paper. We recommend cm × cm paper with mm divisions.

I. Experiment
 
        A. Mapping of Equipotential Lines and Electric Fields

                Electric Fields

The concept of a field surrounding the source of a force is very important in all of physics. In the case of the electric (or Coulomb) force between charged particles, the force is mediated through an electric field The electric fieldin the space surrounding an electrically charged object can be determined at any location  by measuring the force  on a small body carrying a test charge q_o (by convention assumed positive). The electric field is defined as

                The Potential Difference

The difference in the potential at point b from the potential at point a, i.e., V_b -V_{a ,} is given by the line integral

and the unit implied by Eq.(2) for the potential difference, joule/coulomb, is called the volt.

Notice from Eq.(2), that if we choose the points a and b so that we can perform the integral of Eq.(2) along a path whereis perpendicular toandthe potential difference is zero everywhere along the path. Such a path, where the potential is constant, is called an equipotential line, and equipotential lines and electric field lines are mutually perpendicular. The electric fields point from higher equipotential lines to lower equipotential lines.

Fig. 1 shows, as an example, the radial electric field lines which radiate from a central positive point charge, and some of the circular equipotential lines where the potential is a constant. The smaller diameter circles are at increasingly higher potential .

Consider Eq.(2) for a small incremental displacement, for which

where the angle betweenand If = 0, that is, if is in the direction of (or perpendicular to the equipotential line), then the above equation shows we can calculate E approximately from

                Mapping the Electric Fields and Equipotential Lines
 
In the experiment you will perform, a potential difference is established between two electrodes on a sheet of conducting paper which has a uniform electrical resistance (see Fig. 2). A configuration has been painted on the conducting paper with conducting silver paint to serve as positive and negative sources. The system shown produces, in the plane of the conducting paper, an electric field (along which charges flow) and equipotential lines that are somewhat similar to those of a set of static charged point sources.

The experimental setup is shown in Fig. 3.
 

 

• Set up the electric mapping board by inserting the conducting paper with the two electrodes painted on it (Fig. 2). Make sure that there is good contact between the spring clip at the positions of the painted electrode sources.

• Connect the power supply as shown in Fig. 3. The minus side of the power supply (black lead) is connected to one electrode and it becomes the "negative electrode," and likewise, the positive side of the power supply is connected to the other electrode making it the "positive electrode."

 
• (Note. The voltmeter is a device used to measure the potential difference,  between two points. It is calibrated to read directly in volts. When you connect the wires from the voltmeter, make sure the (+) lead of the voltmeter is connected to the point of higher potential and the (-) lead to the point of lower potential. When using a voltmeter, always start by using the highest scale on the meter. )

 
• Connect the voltmeter to the reference probe as shown in Fig. 3, remembering to connect the (-) lead to the negative electrode on the conducting paper and the (+) lead to the reference probe. The switch S is included so that this connection can be broken if desired. The reading on the voltmeter is V_R - V_- where V_- is the potential of the negative electrode. If you define V_- to be 0 volts, then the potential you read is V_R, the potential of the reference probe at its point of contact.

 
• Connect the microammeter to the reference probe and to the movable probe. By tapping the movable probe around on the conducting paper, you can find places where no current flows through the microammeter. At these points on the conducting paper, the reference probe and the movable probe are at the same potential, i.e., V_R = V_M.

Before you turn on your power supply, have your instructor check your circuit.

• Turn on your power supply, setting it to a potential difference of 3 V.

 
• Using the reference probe, find a point on the carbon paper that is at 0.5 V potential above the "negative electrode." Plot and label this point on your graph paper, which represents the conducting paper of Fig. 2.. Leave the reference probe at that position.

 
• Next, open the switch and, by tapping the movable probe along the conducting paper, find points where the current through the microammeter is zero. These are points with no potential difference between the reference probe and movable probe and are called equipotential points. Plot these points directly on your graph paper as you find them on the conducting paper grid.

• Move the reference probe and repeat the procedure for ½ volt increments in the potential of the reference to a maximum of 2.5 V. Also, find one equipotential line very close to one of the "electrodes."

 
• Connect the points on your graph smoothly to form the equipotential lines.

 
• Finally, construct the electric field pattern from the equipotential map, remembering the electric fields and equipotential lines are mutually perpendicular.

• Using Eq.(3),find the numerical value of the electric field for three points on the conducting sheet. Your path increments Dr will be parallel to the electric field (perpendicular to the equipotential line) and probably will be of the order of a centimeter or more. Hence, the E you find will be the value of E approximately half way between the end points of the path increment chosen. Take at least one of these points near the edge of the sheet.

                The Magnetic Field from a Magnetic Dipole

There is a magnetic field in the space surrounding moving charges or electric currents. One of the simplest magnetic field patterns is one that surrounds a magnetic dipole, shown in Fig. 4, which occurs for a simple loop of current. This field pattern looks very similar to that of an electric dipole, shown in Fig. 5.

 

The bar magnet is an example of a magnetic dipole which behaves roughly as though it carries two magnetic opposite charges (or poles) located near the ends. A magnetic compass needle is a tiny bar magnet mounted on a pivot and is used as our test probe of the magnetic field. By convention the positive end is defined to be the North end (i.e. short for "geographic North-seeking" end), and the negative end is called the South end. For a bar magnet, magnetic field lines originate on the North end and terminate on the South end.
 

Because the earth has a molten core in constant rotational motion, it acts like a giant current loop which gives rise to a magnetic dipole field pattern, which we will approximate by a bar magnet. The magnetic field pattern of the earth is shown in Fig. 6.

Notice that as in electrostatics, since like poles repel (N repels N, and S repels S) and unlike poles attract (N attracts S), the geographic north-pole must be an S magnetic pole if the earth is viewed as a large "bar magnet." (To be consistent with the actual configuration, the virtual earth "bar magnet" is shorter than an earth's diameter and not aligned along N-S geographic poles.) A compass needle placed in a magnetic field takes up an orientation parallel to the direction of the horizontal component of the magnetic field vector at that point (the sense of the field is given by the compass arrow with the arrowhead being N or + end). With the compass needle as our probe, we can map the magnetic field lines.

                Experimental Procedure

                Mapping the Combined Field of the Bar Magnet and the Earth

• Attach a large sheet of paper (supplied by the lab instructor) to the table top with masking tape.

 
• Determine the direction of the earth' s field with the compass (keep the bar magnet far away!) and indicate it clearly on the sheet of paper attached to the table.

 
• Place the bar magnet near the center of the paper so that the field near the end is antiparallel to the earth's field. Outline the position of the magnet clearly on the paper. Do not disturb the orientation of the paper or magnet during the experiment. Be careful not to locate the paper above the iron in the tables, if any.

• Placed anywhere on the paper, the compass will now indicate the direction, at that point, of the vector sum of the magnetic fields of the earth and bar magnet. By indicating on the paper with a sharp pencil the ends of the compass needle, then displacing the compass slightly in the direction of the needle, a continuous curve can be traced out to which is everywhere tangent. That is, this " field line" has the same direction asat each point along its length. By tracing a number of such curves from one "pole" of the bar magnet to the other, a schematic picture of the magnetic field configuration emerges. (Trace enough curves to show the general behavior of the field lines in all parts of the paper. If the field appears to be symmetric, you may restrict your actual detailed tracing to one of the regions.) If two people plot at once, make sure that the compasses do not get near each other, or there will be some interaction.

 
• Note that the field lines converge to the position of the "poles"; locate these positions within the bar magnet by projecting lines from various directions and show them on your paper. (Are they at the ends?)

• There are two locations, on the magnet's long axis where the earth's field and the bar magnet's field cancel each other (that is ). This is called a neutral point. Note: The compass points in the direction of the magnetic field, but at a neutral point the magnetic field is zero and the aligning torque on the compass needle vanishes . Magnetic field lines bend away from the neutral point. Find and indicate the position of a neutral point on your sheet.

• From Fig. 6 it is clear the earth's magnetic field is not parallel to the earth's surface at Tallahassee's latitude. Remove the bar magnet from the table. Crudely examine, with a dip compass, the field in the plane perpendicular to the table top which includes the horizontal component of the earth's magnetic field you previously drew on the paper. Estimate the angle with respect to the horizontal that earth's magnetic field enters the earth's surface in Tallahassee.