# Electric field lines bear what geometrical relationship to equipotential

### Electric field direction (video) | Khan Academy In terms of the equipotentials, the relation between \$ and E may be geometrically expressed as follows: the field lines and the equipotentials are everywhere A hollow sphere has a radius of 10 cm and bears a charge of 5 nC. If the sphere is . These are imaginary geometric lines constructed so that the direction of the line, . Figure The apparatus used to map equipotential lines and electric field vectors between This bears some similarity to how the gravitational potential energy was defined. We .. This relationship is complicated by the three dimensions. State and apply the relation between electric force and electric field. .. bears a charge of +Q and the other coin, -Q. Sketch the field lines in the vicinity of the coins. (b) Sketch equipotentials for simple geometries of charged static conductors.

Under certain circumstances, the rules defining these field lines can be used to deduce some general properties of charges and their forces. For example, a property easily deduced from these rules is that a region of space enclosed by a spherically symmetric distribution of charge has zero electric everywhere within that region assuming no additional charges produce electric fields 2 3 inside. Imagine first a spherically symmetric thin shell of positive charge all at a certain distance from the center.

Field lines from the shell would have to be radially outward equally in all directions. If these outward pointing lines continued radially inward beneath the shell, they would have nowhere to end. Hence, the field lines must have ended at the surface charge, and there must be zero field everywhere inside.

Next, suppose the spherically symmetric distribution of charge surrounding the uncharged region is not merely a thin shell. We can nevertheless consider the charge distribution to be divided up into many thin layers each at a different radius.

Each layer contributes its own field lines that end at that layer, producing none of the field lines in the region enclosed by that layer. Then any point in the region of interest is inside all of the thin layers, where all the field lines have ended. We can conclude that the E field is zero at any point within a region surrounded by the spherical distribution of charge. Grass seeds align themselves with the electric field between two charges. The drawing shows the lines of force associated with the electric field between charges.

Figure 1a shows a charge pair with negative charge above and positive charge below. Figure 1b shows two positive charges. It can also be shown that the electric field intensity increases near conducting surfaces that are curved to protrude outward, so that they have a positive curvature.

Curvature is defined as the inverse of the radius. A flat surface has zero curvature. A needle point has a very small radius and a large positive curvature. The larger the curvature of a conducting surface, the greater the field intensity is near the surface. The lightning rod is a pointed conductor. An electrified thundercloud above it attracts charge of the opposite sign to the near end of the rod, but repels charge of the same sign to the far end. The far end is grounded, allowing its charge to escape.

The rod thereby becomes charged by electric induction.

### EQUIPOTENTIAL LINES

The cloud can similarly induce a charge of sign opposite to is own in the ground beneath it. The strongest field results near the point of the lightning rod, and is intense enough to transfer a net charge onto the airborne molecules, thus ionizing them. This produces a glow discharge, in which net electric charge is carried up on the ionized molecules in the air to neutralize part of the charge at the bottom of the cloud before it can produce a lightning bolt.

The electric field representation is not the only way to map how a charge affects the space around it. An equivalent scheme involves the notion of electric potential. The difference in electric potential between two points A and B is defined as the work per unit charge required to move a small positive test charge from point A to point B against the electric force.

For electrostatic forces, it can be shown that this work depends only on the locations of the points A and B and not on the path followed between them in doing the work. Therefore, choosing a convenient point in the region and arbitrarily assigning its electric potential to have some convenient value specifies the electric potential at every other point in the region as the work per unit test charge done to move a test charge between the points.

It is usual to choose either some convenient conductor or else the ground as the reference, and to assign it a potential of zero.

This bears some similarity to how the gravitational potential energy was defined. We could have considered the electric potential energy of a test charge in analogy with the gravitational potential energy by considering the work, not the work per unit charge, done in moving a test charge between two points. But just as the gravitational potential energy itself cannot be used to characterize the gravitational field because it depends on the test mass used, the electric potential energy similarly depends on the test charge used.

But the force and therefore the work to move the test charge from one location to another is proportional to its charge.

## Electric field direction

Thus the work per unit charge, or electric potential difference, is independent of the test charge used as long as the field does not vary in time, so that the electric potential characterizes the electric field itself throughout 4 5 the region of space without regard to the magnitude of the test charge used to probe it.

It is convenient to connect points of equal potential with lines in two dimensional problems; or surfaces in the case of three dimensions. These surfaces are referred to as equipotential surfaces, or equipotentials. If a small test charge is moved so that its direction of motion is always perpendicular to the electric field at each location, then the electric force and the direction of motion at each point are perpendicular. No work is done against the electric force, and the potential at each point traversed is therefore the same.

Hence a path traced out by moving in a direction perpendicular to the electric field at each point is an equipotential. Conversely, if the test charge is moved along an equipotential, there is no change in potential and therefore no work done on the charge by the electric field. For non-zero electric field this can happen only if the charge is being moved perpendicular to the field at each point on such a path.

Therefore, electric field lines and equipotentials always cross at right angles. Figure 2 shows a region of space around a group of charges. The electric lines of force are indicated with solid lines and arrows. The electric field can also be indicated by equipotential lines, shown as dashed lines in the figure. The mapping of a region of space with equipotential lines or, in the case of 3-D space, with equipotential surfaces, provides the same degree of information as by mapping out the electric field itself throughout the region.

Figure 2 Electric field around a group of charges. Lines of force are shown as solid lines. Equipotential lines are shown as dashed lines. The power supply is a source of potential difference which is measured in Newton-meters per Coulomb. When it is connected to the two conductors, a small amount of charge is deposited on each conductor, producing an electric field and maintaining a potential difference, identical to that of the power supply, between the two conductors.

The black paper beneath the conductors is weakly conducting to allow a small current to flow. The digital voltmeter measures the potential difference between the point on the paper where the probe is held and the conductor connected to the other lead of the voltmeter.

Choose the conductor geometry for which you will be mapping the field. Start with a circular conductor on the north terminal post furthest away from you and a horizontal bar on the south terminal nearest you.

Mount these conductor pieces on the brass bolts which protrude from the black-coated paper. Secure the conductors with the brass nuts. Tighten down the nuts to ensure good electrical contact between the conductors and the paper. The positive terminal of the power supply is connected to the red banana jack and the negative terminal to the black banana jack, thereby connecting the power supply to the conductors.

The digital voltmeter measures the potential difference between the two input electrodes. The black ground lead of the digital voltmeter its negative terminal should be connected to the black negative banana jack on the board.

You will use the red positive lead of the digital voltmeter as a probe. Adjust the power supply to maintain the desired Setting the potential difference voltage between the two conductors by following these steps. Connect the red lead of the digital voltmeter to the red banana jack on the board.

The black lead was connected to the black banana jack in the previous step. Adjust the power supply voltage to 6. You may have to change the range setting of the voltmeter to get the proper reading. When this adjustment is completed, disconnect the red lead of the digital voltmeter. The drawing shows the lines of force associated with the electric field between charges.

You might think when several charges are present that the electric field lines from two charges could meet at some location, producing crossed lines of force. But imagine placing a charge where the two lines intersect. Charges are never confused about the direction of the force acting on them, so along which line would the force lie?

In such a case, the electric fields add vectorially at each point, producing a single net E field that lies along one specific line of force, rather than being at the intersection of two lines of force. Thus, it can be seen that none of the lines cross each other. It can also be shown that two field lines never merge to become one. For example, a property easily deduced from these rules is that a region of space enclosed by a spherically symmetric distribution of charge has zero electric field everywhere within that region assuming no additional charges produce electric fields inside.

Imagine first a spherically symmetric thin shell of positive charge all at a certain distance from the center. Field lines from the shell would have to be radially Figure 4. Electric field around a group of outward equally in all directions. Lines of force are shown as solid outward pointing lines continued radially lines. Equipotential lines are shown as dashed inward beneath the shell, they would have nowhere to end.

Hence, the field lines must lines. Next, suppose the spherically symmetric distribution of charge surrounding the uncharged region is not merely a thin shell.

We can nevertheless consider the charge distribution to be divided up into many thin layers each at a different radius. Each layer contributes its own field lines that end at that layer, producing none of the field lines in the region enclosed by that layer.

Then any point in the region of interest is inside all of the thin layers, where all the field lines have ended. We can conclude that the E field is zero at any point within a region surrounded by the spherical distribution of charge. It can also be shown that the electric field intensity increases near conducting surfaces that are curved to protrude outward, so that they have a positive curvature.

Curvature is defined as the inverse of the radius. A flat surface has zero curvature. A needle point has a very small radius and a large positive curvature. The larger the curvature of a conducting surface, the greater the field intensity is near the surface.

Checkpoint What are three properties of electric lines of force? Why do electric lines of force never cross? Checkpoint How do the electric lines of force represent an increasing field intensity? The apparatus used to map equipotential lines and electric field vectors between two electrodes.

Historical Aside The lightning rod is a pointed conductor. An electrified thundercloud above it attracts charge of the opposite sign to the near end of the rod, but repels charge of the same sign to the far end. The far end is grounded, allowing its charge to move across the earth. The rod thereby becomes charged by electric induction. The cloud can similarly induce a charge of sign opposite to is own in the ground beneath it. The strongest field results near the point of the lightning rod, and is intense enough to transfer a net charge onto the airborne molecules, thus ionizing them.

This produces a glow discharge, in which net electric charge is carried up on the ionized molecules in the air to neutralize part of the charge at the bottom of the cloud before it can produce a lightning bolt. An equivalent scheme involves the notion of electric potential. The difference in electric potential between two points A and B is defined as the work per unit charge required to move a small positive test charge from point A to point B against the electric force.

For electrostatic forces, it can be shown that this work depends only on the locations of the points A and B and not on the path followed between them in doing the work. Therefore, choosing a convenient point in the region and arbitrarily assigning its electric potential to have some convenient value specifies the electric potential at every other point in the region as the work per unit test charge done to move a test charge between the points.

It is usual to choose either some convenient conductor or else the ground as the reference, and to assign it a potential of zero. General Information This bears some similarity to how the gravitational potential energy was defined. But just as the gravitational potential energy itself cannot be used to characterize the gravitational field because it depends on the test mass used, the electric potential energy similarly depends on the test charge used.

But the force and therefore the work to move the test charge from one location to another is proportional to its charge. Thus the work per unit charge, or electric potential difference, is independent of the test charge used as long as the field does not vary in time, so that the electric potential characterizes the electric field itself throughout the region of space without regard to the magnitude of the test charge used to probe it.

It is convenient to connect points of equal potential with lines in two dimensional problems; or surfaces in the case of three dimensions. These lines are called equipotential lines; these surfaces are called equipotential surfaces; volumes, surfaces, or lines whose points all have the same electric potential are called equipotentials.

• EQUIPOTENTIAL LINES. Electric fields

If a small test charge is moved so that its direction of motion is always perpendicular to the electric field at each location, then the electric force and the direction of motion at each point are perpendicular. No work is done against the electric force, and the potential at each point traversed is therefore the same.

Hence a path traced out by moving in a direction perpendicular to the electric field at each point is an equipotential. Conversely, if the test charge is moved along an equipotential, there is no change in potential and therefore no work done on the charge by the electric field.

For non-zero electric field this can happen only if the charge is being moved perpendicular to the field at each point on such a path. Therefore, electric field lines and equipotentials always cross at right angles. The electric lines of force are indicated with solid lines and arrows. The mapping of a region of space with equipotential lines or, in the case of 3-D space, with equipotential surfaces, provides the same degree of information as by mapping out the electric field itself throughout the region.

Checkpoint Are the electric field representation and the equipotential line representation equivalent in terms of how much information they contain about the electric field? We define U x, y, z in such a way that total energy is conserved. Checkpoint What are the units of potential difference? What are units of electric field? Mapping equipotentials between oppositely charged conductors The equipotential apparatus is shown in Figure 4. The power supply is a source of potential difference work per unit charge measured in Volts V. When it is connected to the two conductors, a small amount of charge is deposited on each conductor, producing an electric field and maintaining a potential difference, identical to that of the power supply, between the two conductors.

The black paper beneath the conductors is weakly conducting to allow a small current to flow. The digital voltmeter measures the potential difference between the point on the paper where the probe is held and the conductor connected to the other lead of the voltmeter.

The voltmeter is efficient at determining potential difference using a very small but nonzero current. Remove the electrodes left behind by the last class. Choose the conductor geometry for which you will be mapping the field.

Start with a circular conductor on the terminal post furthest away from you and a horizontal bar on the terminal nearest you. Wipe away any eraser crumbs from the area of the electrodes.

Mount these conductor pieces on the brass bolts which protrude from the black-coated paper. Each electrode has a raised lip around its edge on one side. This side must face down so that the raised lip makes good electrical contact with the black paper. Secure the conductors with the brass nuts. Tighten down the nuts well to ensure good electrical contact between the conductors and the paper. The banana jack away from you is red and the jack close to you is black.

The positive terminal of the power supply is connected to the red banana jack and the negative terminal to the black banana jack. These jacks are connected to the bolt holding the round electrode and to the center bolt holding the bar, respectively, using wires under the apparatus.

The digital voltmeter measures the potential difference between the two input banana jacks. The black terminal of the digital voltmeter its negative terminal should be connected to the black negative banana jack on the board and to the negative power supply terminal.

## EQUIPOTENTIAL LINES

You will use the red positive lead of the digital voltmeter connected to its positive terminal as an electric potential probe to map out V x, y in the plane of the paper. Choose a few points at random on the black paper and place the red voltmeter probe lightly at these points in turn. Note this observation in your Data. Touch the red potential probe to the round electrode and hold it there.

Rotate the Voltage adjustments on the power supply. The Coarse adjustment changes the voltage fast and the Fine adjustment changes the voltage slowly. Adjust the power supply voltage to 6. The voltmeter has more digits than you need for this experiment, 5. You may have to change the range setting of the voltmeter to get the proper reading. Note that all points on the round electrode have the same voltage. The electrodes are equipotential volumes and their surfaces are equipotential surfaces. When this adjustment is completed, remove the probe from the round electrode.

Note the voltage of each of the electrodes in your Data. You are now ready to take data. The goal is to locate points at each desired potential in order to trace out the corresponding equipotential line. Suppose, for example, you want to find an equipotential at 5. Lightly place the red probe on the surface of the black paper and gently move it around until the digital voltmeter reads 5.

This point is then at a potential of 5 Volts above that of the bar conductor. We need to determine the x, y coordinates of this point so that we can plot it on the graph paper. The bar is inscribed with marks at every two centimeters.

One side of the bar is inscribed every two millimeters. These marks can be used as our x-coordinates. We also have a plastic right triangle and a ruler that we can use to determine the x, y-coordinates. Plot the point on the graph paper and draw a box, triangle, diamond, star, etc.