Why do equipotential lines curve

The total potential difference is , so of the distance between the plates will be the distance between potential differences. The distance between the plates is , so there will be between potential differences. You have now seen a numerical calculation of the locations of equipotentials between two charged parallel plates. In Example 3. Given that a conducting sphere in electrostatic equilibrium is a spherical equipotential surface, we should expect that we could replace one of the surfaces in Example 3.

Inside will be rather different, however. To investigate this, consider the isolated conducting sphere of Figure 3. To find the electric field both inside and outside the sphere, note that the sphere is isolated, so its surface change distribution and the electric field of that distribution are spherically symmetric.

We can therefore represent the field as. Since is constant and on the sphere,. If , encloses the conductor so. As expected, in the region , the electric field due to a charge placed on an isolated conducting sphere of radius is identical to the electric field of a point charge located at the centre of the sphere. To find the electric potential inside and outside the sphere, note that for the potential must be the same as that of an isolated point charge located at ,.

For , , so is constant in this region. Since ,. The spheres are sufficiently separated so that each can be treated as if it were isolated aside from the wire. Note that the connection by the wire means that this entire system must be an equipotential. We have just seen that the electrical potential at the surface of an isolated, charged conducting sphere of radius is. Now, the spheres are connected by a conductor and are therefore at the same potential; hence.

The net charge on a conducting sphere and its surface charge density are related by. Substituting this equation into the previous one, we find.

Obviously, two spheres connected by a thin wire do not constitute a typical conductor with a variable radius of curvature. Nevertheless, this result does at least provide a qualitative idea of how charge density varies over the surface of a conductor.

The equation indicates that where the radius of curvature is large points and in Figure 3. Similarly, the charges tend to be denser where the curvature of the surface is greater, as demonstrated by the charge distribution on oddly shaped metal Figure 3.

The surface charge density is higher at locations with a small radius of curvature than at locations with a large radius of curvature. A practical application of this phenomenon is the lightning rod , which is simply a grounded metal rod with a sharp end pointing upward.

As positive charge accumulates in the ground due to a negatively charged cloud overhead, the electric field around the sharp point gets very large. When the field reaches a value of approximately the dielectric strength of the air , the free ions in the air are accelerated to such high energies that their collisions with air molecules actually ionize the molecules.

The resulting free electrons in the air then flow through the rod to Earth, thereby neutralizing some of the positive charge. This keeps the electric field between the cloud and the ground from getting large enough to produce a lightning bolt in the region around the rod.

An important application of electric fields and equipotential lines involves the heart. The heart relies on electrical signals to maintain its rhythm. The movement of electrical signals causes the chambers of the heart to contract and relax. When a person has a heart attack, the movement of these electrical signals may be disturbed. An artificial pacemaker and a defibrillator can be used to initiate the rhythm of electrical signals. The equipotential lines around the heart, the thoracic region, and the axis of the heart are useful ways of monitoring the structure and functions of the heart.

An electrocardiogram ECG measures the small electric signals being generated during the activity of the heart. Play around with this simulation to move point charges around on the playing field and then view the electric field, voltages, equipotential lines, and more.

Skip to content By the end of this section, you will be able to: Define equipotential surfaces and equipotential lines Explain the relationship between equipotential lines and electric field lines Map equipotential lines for one or two point charges Describe the potential of a conductor Compare and contrast equipotential lines and elevation lines on topographic maps.

The potential is the same along each equipotential line, meaning that no work is required to move a charge anywhere along one of those lines. Work is needed to move a charge from one equipotential line to another. Move point charges around on the playing field and then view the electric field, voltages, equipotential lines, and more. Figure 5. The electric field near two equal positive charges is directed away from each of the charges. Figure 9. A charged insulating rod such as might be used in a classroom demonstration.

Figure Skip to main content. Electric Potential and Electric Field. Search for:. Equipotential Lines Learning Objectives By the end of this section, you will be able to: Explain equipotential lines and equipotential surfaces.

Describe the action of grounding an electrical appliance. Compare electric field and equipotential lines. Grounding A conductor can be fixed at zero volts by connecting it to the earth with a good conductor—a process called grounding. PhET Explorations: Charges and Fields Move point charges around on the playing field and then view the electric field, voltages, equipotential lines, and more.

Click to run the simulation. Conceptual Questions What is an equipotential line? What is an equipotential surface? Explain in your own words why equipotential lines and surfaces must be perpendicular to electric field lines.

Can different equipotential lines cross? Indicate the direction of increasing potential. Sketch the equipotential lines for the two equal positive charges shown in Figure 5.

Figure 6. The electric field near two charges. Figure 7. A negatively charged conductor. Figure 8. Licenses and Attributions. Does the location of your reference point matter? A metallic sphere of radius 2. The metallic sphere stands on an insulated stand and is surrounded by a larger metallic spherical shell, of inner radius 5. Now, a charge of is placed on the inside of the spherical shell, which spreads out uniformly on the inside surface of the shell.

If potential is zero at infinity, what is the potential of a the spherical shell, b the sphere, c the space between the two, d inside the sphere, and e outside the shell? Two large charged plates of charge density face each other at a separation of 5. A long cylinder of aluminum of radius R meters is charged so that it has a uniform charge per unit length on its surface of. Two parallel plates 10 cm on a side are given equal and opposite charges of magnitude The plates are 1.

What is the potential difference between the plates? The surface charge density on a long straight metallic pipe is. What is the electric potential outside and inside the pipe? Assume the pipe has a diameter of 2 a. Concentric conducting spherical shells carry charges Q and — Q , respectively.

The inner shell has negligible thickness. What is the potential difference between the shells? In the region , and E is zero elsewhere; hence, the potential difference is.

Shown below are two concentric spherical shells of negligible thicknesses and radii and The inner and outer shell carry net charges and respectively, where both and are positive. What is the electric potential in the regions a b and c. A solid cylindrical conductor of radius a is surrounded by a concentric cylindrical shell of inner radius b.

The solid cylinder and the shell carry charges Q and — Q , respectively. Assuming that the length L of both conductors is much greater than a or b , what is the potential difference between the two conductors? From previous results , note that b is a very convenient location to define the zero level of potential:. Skip to content Electric Potential. Learning Objectives By the end of this section, you will be able to: Define equipotential surfaces and equipotential lines Explain the relationship between equipotential lines and electric field lines Map equipotential lines for one or two point charges Describe the potential of a conductor Compare and contrast equipotential lines and elevation lines on topographic maps.

An isolated point charge Q with its electric field lines in red and equipotential lines in black. The potential is the same along each equipotential line, meaning that no work is required to move a charge anywhere along one of those lines. Work is needed to move a charge from one equipotential line to another.

Equipotential lines are perpendicular to electric field lines in every case. For a three-dimensional version, explore the first media link. The electric field lines and equipotential lines for two equal but opposite charges. The equipotential lines can be drawn by making them perpendicular to the electric field lines, if those are known.

Note that the potential is greatest most positive near the positive charge and least most negative near the negative charge. Electric potential map of two opposite charges of equal magnitude on conducting spheres.

The potential is negative near the negative charge and positive near the positive charge. A cross-section of the electric potential map of two opposite charges of equal magnitude. The electric field and equipotential lines between two metal plates.

Note that the electric field is perpendicular to the equipotentials and hence normal to the plates at their surface as well as in the center of the region between them. A topographical map along a ridge has roughly parallel elevation lines, similar to the equipotential lines in Figure. Lines that are close together indicate very steep terrain. Notice the top of the tower has the same shape as the center of the topographical map.

Their locations are: ; ; ;. The electric field between oppositely charged parallel plates. A portion is released at the positive plate. Distribution of Charges on Conductors In Figure with a point charge, we found that the equipotential surfaces were in the form of spheres, with the point charge at the center.

An isolated conducting sphere. Two conducting spheres are connected by a thin conducting wire. The surface charge density and the electric field of a conductor are greater at regions with smaller radii of curvature. Summary An equipotential surface is the collection of points in space that are all at the same potential. Equipotential lines are the two-dimensional representation of equipotential surfaces.

Equipotential surfaces are always perpendicular to electric field lines. Conductors in static equilibrium are equipotential surfaces. Topographic maps may be thought of as showing gravitational equipotential lines. Conceptual Questions If two points are at the same potential, are there any electric field lines connecting them? No; it might not be at electrostatic equilibrium. Can a positively charged conductor be at a negative potential? Can equipotential surfaces intersect? Problems Two very large metal plates are placed 2.