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Electric Charge and Electric Field

8 Conductors and Electric Fields in Static Equilibrium

Learning Objectives

  • List the three properties of a conductor in electrostatic equilibrium.
  • Explain the effect of an electric field on free charges in a conductor.
  • Explain why no electric field may exist inside a conductor.
  • Describe the electric field surrounding Earth.
  • Explain what happens to an electric field applied to an irregular conductor.
  • Describe how a lightning rod works.
  • Explain how a metal car may protect passengers inside from the dangerous electric fields caused by a downed line touching the car.

Conductors are materials (such as metals) that contain free charges—charges that can move easily through the material. In most metals, these mobile charges are electrons. When excess charge is placed on a conductor, or when a conductor is placed in an external electric field, the free charges respond quickly and redistribute until the system reaches a steady state called electrostatic equilibrium.

Figure 8.1 shows how an applied electric field affects free charges in a conductor. The free charges move until the electric field at the surface is perpendicular to the conductor. In electrostatic equilibrium, there can be no component of the electric field parallel to the surface—because any parallel component would push charges along the surface and charge motion would continue (meaning the system would not be “at rest” electrostatically).

In the diagram, a positive free charge is shown for clarity. In metals the mobile charges are negative (electrons), but the physics is the same: the motion of a positive charge in one direction is equivalent to the motion of a negative charge in the opposite direction.

In part a, an electric field E exists at some angle with the horizontal applied on a conductor. One component of this field E parallel is along x axis represented by a vector arrow and other E perpendicular, is along y axis represented by a vector arrow. Charge inside the conductor moves along x axis so the force acting on it is F parallel, which is equal to q multiplied by E parallel. In part b, a charge is shown inside the conductor and electric field is represented by a vector arrow pointing upward starting from the surface of the conductor.
Figure 8.1: When an electric field [latex]\mathbf{E}[/latex] is applied to a conductor, free charges inside the conductor move until the field is perpendicular to the surface. (a) The electric field has both parallel and perpendicular components. The parallel component, [latex]\mathbf{E}_{\parallel}[/latex], exerts a force [latex]\mathbf{F}_{\parallel}[/latex] on a charge [latex]q[/latex], causing charges to move until the parallel component is eliminated (so [latex]\mathbf{F}_{\parallel}=0[/latex]). (b) The resulting electric field is perpendicular to the surface; charges have redistributed to the surface, and electrostatic forces are in equilibrium.

A neutral conductor placed in an external electric field becomes polarized: mobile charges shift slightly so that one side becomes more negative and the opposite side becomes more positive. Figure 8.2 shows the result of placing a neutral conductor in an originally uniform electric field. The field becomes distorted around the conductor (often stronger near the surface), but it disappears inside the conductor once electrostatic equilibrium is reached.

A spherical conductor is placed in the external electric field. The field lines are shown running from left to right. The field lines enter and leave the conductor at right angles. Negative charges accumulate on the left surface of the conductor and positive charges accumulate on the right surface of the conductor.
Figure 8.2: A spherical conductor in static equilibrium within an originally uniform electric field. Free charges move and polarize the conductor until the field lines are perpendicular to the surface. Field lines end on excess negative charge on one region of the surface and begin again on excess positive charge on the opposite side. No electric field exists inside the conductor; if it did, free charges would keep moving until the internal field was canceled.

Misconception Alert: Electric Field inside a Conductor

Excess charges placed on a spherical conductor repel one another and move until they are evenly distributed on the outer surface, as shown in Figure 8.3. The excess charge ends up on the surface because, at equilibrium, the electric field inside the conducting material must be zero.

A useful consequence is that the electric field outside a charged spherical conductor is the same as if all of its excess charge were concentrated at a single point at the center (as far as points outside the sphere are concerned). This “acts like a point charge” idea is extremely helpful in modeling charged objects in physics and in biomedical technology (for example, understanding how charge distributions influence fields near electrodes).

A positively charged sphere is shown and positive charges are distributed all over the surface. Electric field lines emanate from the sphere in the space shown by the vector arrow pointing outward.
Figure 8.3: Mutual repulsion of excess charges on a spherical conductor spreads them uniformly over the surface. The resulting electric field is perpendicular to the surface and zero inside the conductor. Outside the conductor, the field is identical to that of a point charge at the center equal to the total excess charge.

Properties of a Conductor in Electrostatic Equilibrium

  1. The electric field is zero inside a conductor.
  2. Just outside a conductor, the electric field lines are perpendicular to its surface, ending on negative charges or beginning on positive charges on the surface.
  3. Any excess charge resides entirely on the surface (or surfaces) of a conductor.

These properties summarize how conductors behave in electrostatic equilibrium and can be used to analyze many real situations—including biomedical instrumentation where metal electrodes and shields are designed to control electric fields and reduce unwanted interference.

One application is creating a very uniform electric field. Consider two large metal plates with equal and opposite charges, as shown in Figure 8.4. In electrostatic equilibrium, excess charge spreads nearly uniformly over the facing surfaces, producing electric field lines that are straight, uniformly spaced (uniform strength), and perpendicular to the plates (uniform direction), except near the edges. Edge effects are less important when the plates are close compared with their size.

Two charged metal plates are shown. The lower plate has negative charge and the upper plate has positive charge. The electric field lines start from positive plate and enter the negative plate represented by arrows.
Figure 8.4: Two metal plates with equal but opposite excess charge produce a nearly uniform electric field between them (except near the edges). Uniform fields are useful when we want charges to experience a predictable force—an idea used in many devices that accelerate or steer charged particles.

Earth’s Electric Field

A roughly uniform electric field surrounds Earth with a typical fair-weather magnitude of about 150 N/C, directed downward (toward Earth’s surface). The magnitude increases slightly closer to the ground. What produces this field? Roughly 100 km above Earth’s surface is a region containing many charged particles called the ionosphere. In fair weather, the ionosphere tends to be positively charged relative to Earth’s surface, which is largely negative, producing the field shown in Figure 8.5(a).

In storm conditions, cloud charge separation can create much stronger local electric fields, and the field direction can be different from the fair-weather case (Figure 8.5(b)). The precise charge distribution depends on the storm structure and local conditions.

If the electric field becomes large enough, the insulating properties of air break down and air becomes conducting. For air, breakdown occurs around

[latex]E \approx 3\times 10^{6}\ \text{N/C}[/latex]

During breakdown, air molecules are ionized, electrons and ions move rapidly, and charge can discharge in the form of lightning or corona discharge.

In part a, a child is flying a kite with two men in an open field on a bright sunny day. In part b, lightning appears over a body of water in stormy weather.
Figure 8.5: Earth’s electric field. (a) In fair weather, Earth and the ionosphere (a layer of charged particles) behave like conductors separated by air, producing a downward electric field of about 150 N/C. (credit: D. H. Parks) (b) In storms, cloud charge separation can create larger local fields. If the field becomes high enough, air breaks down and lightning can occur. (credit: Jan-Joost Verhoef)

Electric Fields on Uneven Surfaces

So far we have emphasized smooth, symmetric conductors. What happens if a conductor has sharp corners or a pointed tip? Excess charge on a nonuniform conductor becomes more concentrated at regions of greatest curvature (the sharpest points). In addition, charge is more likely to leak off (or be pulled onto) the conductor at those sharp regions because the electric field can become very large there.

To see how and why, consider the charged conductor in Figure 8.6. Electrostatic repulsion between like charges spreads charge out, but it spreads charge most effectively where the surface is flatter. The reason is geometric: even if the forces between pairs of identical charges are similar, the component of the force parallel to the surface differs from place to place. It is the component parallel to the surface, [latex]\mathbf{F}_{\parallel}[/latex], that can “push” charges along the surface after they have reached it.

The same concentration effect occurs when an external electric field is applied to an irregular conductor (Figure 8.6(c)). Because the field lines must be perpendicular to the surface in electrostatic equilibrium, they crowd together near sharp points, indicating a stronger electric field in those regions.

In part a, a conductor is shown with the unsymmetrical shape. The identical pair of charges at opposite ends on the conductor have similar components of forces represented by arrows. In part b, the unsymmetrical object has positive charge on its surface. The electric field lines are shown emerging perpendicular from the surface of the conductor represented by vector arrow. In part c, the field lines in and around the conductor running from left to right is shown. The left surface of the conductor has negative charge and the right surface has positive charge. The field lines enter and leave the conductor at right angles.
Figure 8.6: Excess charge on a nonuniform conductor becomes most concentrated at the region of greatest curvature (the sharpest end). (a) Forces between identical pairs of charges can be comparable, but the components parallel to the surface differ. (b) Where [latex]\mathbf{F}_{\parallel}[/latex] is smaller, charges are less able to spread out, so they remain closer together, producing a stronger local electric field. (c) A neutral conductor in an originally uniform electric field becomes polarized, with the most concentrated induced charge at the most pointed end.

Applications of Conductors

On a sharply curved surface, charges can become so concentrated that the electric field at the tip becomes extremely large (Figure 8.7). If the field is strong enough, it can remove charge from the conductor into the surrounding air (often by ionizing the nearby air). This effect is useful in some technologies and is central to how lightning protection works.

Lightning rods work best when they are pointed. Charges in storm clouds induce opposite charge on objects below (including buildings). A pointed rod produces a very strong local electric field that can slowly leak charge into the air (corona discharge), reducing the buildup of large charge differences. In many cases, this can lower the probability of a sudden, damaging lightning strike. When properly installed, lightning rods also provide a preferred low-resistance path to ground if a strike does occur.

In other situations, we want to prevent charge from leaking into the air. Then the conductor should be smooth and have a large radius of curvature (no sharp points). Smooth components are used on high-voltage transmission systems to reduce unwanted charge leakage and prevent corona discharge. (See Figure 8.8.)

Another application is a Faraday cage, a conducting enclosure. In electrostatic equilibrium, excess charge resides on the outside surface, and the electric field inside the enclosed region is (ideally) zero. Faraday cages are used to shield sensitive electronics from external fields and noise. A biomedical example is shielding in instrumentation that measures tiny bioelectric signals (such as ECG or EEG), where external electromagnetic interference could otherwise overwhelm the signal of interest.

This same shielding concept helps explain why a metal car can protect passengers during an electrical hazard. During a lightning strike nearby, or if a downed power line touches the car, charge tends to travel along the outside of the metal body. The electric field inside the passenger compartment is much smaller (ideally near zero), provided occupants remain inside and do not simultaneously touch the metal exterior and the ground. This is why safety guidance emphasizes staying in the vehicle until help arrives and the hazard is removed.

A cone shaped positively charged conductor is shown where most of the positive charges are accumulated at the tip. The field lines represented by arrows emerge at right angles from the surface of the conductor in outward direction. The density of field lines is greater at the tip of the cone than at other surfaces.
Figure 8.7: A sharply pointed conductor concentrates charge at the tip, producing a very strong local electric field. If the field is large enough, it can transfer charge between the conductor and the surrounding air. Lightning rods use this effect: a pointed tip helps reduce dangerous charge buildup on structures and can provide a preferred path for discharge.
In part a, a lightning rod is shown on the roof of a house. In part b, a person is touching the metal sphere of the Van De Graaff and his hair is standing up.
Figure 8.8: (a) A lightning rod is pointed to facilitate controlled charge transfer between the rod and the air, helping reduce extreme charge buildup on a structure. (credit: Romaine, Wikimedia Commons) (b) A Van de Graaff generator has a smooth, rounded surface (large radius of curvature) to reduce charge leakage and allow a large voltage to build up. The mutual repulsion of like charges is visible in the person’s hair. (credit: Jon ‘ShakataGaNai’ Davis/Wikimedia Commons)

Section Summary

  • A conductor allows free charges to move about within it.
  • The electrical forces around a conductor will cause free charges to move around inside the conductor until static equilibrium is reached.
  • Any excess charge will collect along the surface of a conductor.
  • Conductors with sharp corners or points will collect more charge at those points.
  • A lightning rod is a conductor with sharply pointed ends that collect excess charge on the building caused by an electrical storm and allow it to dissipate back into the air.
  • Electrical storms result when the electrical field of Earth’s surface in certain locations becomes more strongly charged, due to changes in the insulating effect of the air.
  • A Faraday cage acts like a shield around an object, preventing electric charge from penetrating inside.

Conceptual Questions

  1. Is the object in Figure 8.9 a conductor or an insulator? Justify your answer.
    External field lines entering the object from one end and emerging from another are shown by lines.
    Figure 8.9
  2. If the electric field lines in the figure above were perpendicular to the object, would it necessarily be a conductor? Explain.
  3. The discussion of the electric field between two parallel conducting plates, in this module states that edge effects are less important if the plates are close together. What does close mean? That is, is the actual plate separation crucial, or is the ratio of plate separation to plate area crucial?
  4. Would the self-created electric field at the end of a pointed conductor, such as a lightning rod, remove positive or negative charge from the conductor? Would the same sign charge be removed from a neutral pointed conductor by the application of a similar externally created electric field? (The answers to both questions have implications for charge transfer utilizing points.)
  5. Why is a golfer with a metal club over her shoulder vulnerable to lightning in an open fairway? Would she be any safer under a tree?
  6. Can the belt of a Van de Graaff accelerator be a conductor? Explain.
  7. Are you relatively safe from lightning inside an automobile? Give two reasons.
  8. Discuss pros and cons of a lightning rod being grounded versus simply being attached to a building.
  9. Using the symmetry of the arrangement, show that the net Coulomb force on the charge [latex]q[/latex] at the center of the square below Figure 8.10 is zero if the charges on the four corners are exactly equal.
    Four point charges, one is q a, second is q b, third is q c, and fourth is q d, lie on the corners of a square. q is located at its center.
    Figure 8.10: Four point charges [latex]{q}_{a}[/latex], [latex]{q}_{b}[/latex], [latex]{q}_{c}[/latex], and [latex]{q}_{d}[/latex] lie on the corners of a square and [latex]q[/latex] is located at its center.
  10. (a) Using the symmetry of the arrangement, show that the electric field at the center of the square in Figure 8.10 is zero if the charges on the four corners are exactly equal. (b) Show that this is also true for any combination of charges in which [latex]{q}_{a}={q}_{d}[/latex] and [latex]{q}_{b}={q}_{c}[/latex]
  11. (a) What is the direction of the total Coulomb force on [latex]q[/latex] in Figure 8.10 if [latex]q[/latex] is negative, [latex]{q}_{a}={q}_{c}[/latex] and both are negative, and [latex]{q}_{b}={q}_{c}[/latex] and both are positive? (b) What is the direction of the electric field at the center of the square in this situation?
  12. Considering Figure 8.10, suppose that [latex]{q}_{a}={q}_{d}[/latex] and [latex]{q}_{b}={q}_{c}[/latex]. First show that [latex]q[/latex] is in static equilibrium. (You may neglect the gravitational force.) Then discuss whether the equilibrium is stable or unstable, noting that this may depend on the signs of the charges and the direction of displacement of [latex]q[/latex] from the center of the square.
  13. If [latex]{q}_{a}=0[/latex] in Figure 8.10, under what conditions will there be no net Coulomb force on [latex]q[/latex]?
  14. In regions of low humidity, one develops a special “grip” when opening car doors, or touching metal door knobs. This involves placing as much of the hand on the device as possible, not just the ends of one’s fingers. Discuss the induced charge and explain why this is done.
  15. Tollbooth stations on roadways and bridges usually have a piece of wire stuck in the pavement before them that will touch a car as it approaches. Why is this done?
  16. Suppose a woman carries an excess charge. To maintain her charged status can she be standing on ground wearing just any pair of shoes? How would you discharge her? What are the consequences if she simply walks away?

Problems & Exercises

  1. Sketch the electric field lines in the vicinity of the conductor in Figure 8.11 given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
    A oblong-shaped conductor.
    Figure 8.11
  2. Sketch the electric field lines in the vicinity of the conductor in Figure 8.12 given the field was originally uniform and parallel to the object’s long axis. Is the resulting field small near the long side of the object?
    A oblong-shaped conductor.
    Figure 8.12
  3. Sketch the electric field between the two conducting plates shown in Figure 8.13, given the top plate is positive and an equal amount of negative charge is on the bottom plate. Be certain to indicate the distribution of charge on the plates.
    Two plates are shown; one is in horizontal direction and other is above the first plate with some inclination.
    Figure 8.13
  4. Sketch the electric field lines in the vicinity of the charged insulator in Figure 8.14 noting its nonuniform charge distribution.
    A positively charged rod with a concentration of positive charges near the top and a few in the middle.
    Figure 8.14 A charged insulating rod such as might be used in a classroom demonstration.
  5. What is the force on the charge located at [latex]x=8.00 cm[/latex] in Figure 8.15 (a) given that [latex]q=1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]?
    Three point charges are shown on the scaling line. First charge plus q is at three point zero, second charge minus two q is at eight point zero, and third charge plus q is eleven point zero centimeters along the x axis. Four charges are placed on a scaling line. First is minus two q at one point zero, second is plus q at five point zero, third is plus three q is at eight point zero, and fourth is minus q placed at fourteen point zero centimeter along the x axis.
    Figure 8.15 (a) Point charges located at 3.00, 8.00, and 11.0 cm along the x-axis. (b) Point charges located at 1.00, 5.00, 8.00, and 14.0 cm along the x-axis.
  6. (a) Find the total electric field at [latex]x=1.00 cm[/latex] in  Figure 8.15 (b) given that [latex]q=5.00 nC[/latex]. (b) Find the total electric field at [latex]x=11.00 cm[/latex] in Figure 8.15 (b). (c) If the charges are allowed to move and eventually be brought to rest by friction, what will the final charge configuration be? (That is, will there be a single charge, double charge, etc., and what will its value(s) be?)
  7. (a) Find the electric field at [latex]x=5.00 cm[/latex] in Figure 8.15 (a), given that [latex]q=1.00\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]. (b) At what position between 3.00 and 8.00 cm is the total electric field the same as that for [latex]–2q[/latex] alone? (c) Can the electric field be zero anywhere between 0.00 and 8.00 cm? (d) At very large positive or negative values of x, the electric field approaches zero in both (a) and (b). In which does it most rapidly approach zero and why? (e) At what position to the right of 11.0 cm is the total electric field zero, other than at infinity? (Hint: A graphing calculator can yield considerable insight in this problem.)
  8. (a) Find the total Coulomb force on a charge of 2.00 nC located at [latex]x=4.00 cm[/latex] in Figure 8.15 (b), given that [latex]q=1.00\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]. (b) Find the x-position at which the electric field is zero in Figure 8.15 (b).
  9. Using the symmetry of the arrangement, determine the direction of the force on [latex]q[/latex] in the figure below, given that [latex]{q}_{a}={q}_{b}\text{=+}7\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex] and [latex]{q}_{c}={q}_{d}=-7\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]. (b) Calculate the magnitude of the force on the charge [latex]q[/latex], given that the square is 10.0 cm on a side and [latex]q=2\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex].
    Four point charges, one is q a, second is q b, third is q c, and fourth is q d, lie on the corners of a square. q is located at its center.
    Figure 8.16
  10. (a) Using the symmetry of the arrangement, determine the direction of the electric field at the center of the square in Figure 8.16, given that [latex]{q}_{a}={q}_{b}=-1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex] and [latex]{q}_{c}={q}_{d}\text{=+}1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]. (b) Calculate the magnitude of the electric field at the location of [latex]q[/latex], given that the square is 5.00 cm on a side.
  11. Find the electric field at the location of [latex]{q}_{a}[/latex] in Figure 8.16 given that [latex]{q}_{b}={q}_{c}={q}_{d}\text{=+}2\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex], [latex]q=-1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex], and the square is 20.0 cm on a side.
  12. Find the total Coulomb force on the charge [latex]q[/latex] in Figure 8.16, given that [latex]q=1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex], [latex]{q}_{a}=2\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex], [latex]{q}_{b}=-3\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex], [latex]{q}_{c}=-4\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex], and [latex]{q}_{d}\text{=+}1\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{μC}[/latex]. The square is 50.0 cm on a side.
  13. (a) Find the electric field at the location of [latex]{q}_{a}[/latex] in Figure 8.16, given that [latex]{q}_{\text{b}}=+10.00\phantom{\rule{0.25em}{0ex}}\mu \text{C}[/latex] and [latex]{q}_{\text{c}}=–5.00\phantom{\rule{0.25em}{0ex}}\mu \text{C}[/latex]. (b) What is the force on [latex]{q}_{a}[/latex], given that [latex]{q}_{\text{a}}=+1.50\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex]?
    Three point charges located at the corners of an equilateral triangle.
    Figure 8.17 Point charges located at the corners of an equilateral triangle 25.0 cm on a side.
  14. (a) Find the electric field at the center of the triangular configuration of charges in Figure 8.17, given that [latex]{q}_{a}\text{=+}2\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex], [latex]{q}_{b}=-8\text{.}\text{00}\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex], and [latex]{q}_{c}\text{=+}1\text{.}\text{50}\phantom{\rule{0.25em}{0ex}}\text{nC}[/latex]. (b) Is there any combination of charges, other than [latex]{q}_{a}={q}_{b}={q}_{c}[/latex], that will produce a zero strength electric field at the center of the triangular configuration?

Glossary

conductor
an object with properties that allow charges to move about freely within it
free charge
an electrical charge (either positive or negative) which can move about separately from its base molecule
electrostatic equilibrium
an electrostatically balanced state in which all free electrical charges have stopped moving about
polarized
a state in which the positive and negative charges within an object have collected in separate locations
ionosphere
a layer of charged particles located around 100 km above the surface of Earth, which is responsible for a range of phenomena including the electric field surrounding Earth
Faraday cage
a metal shield which prevents electric charge from penetrating its surface

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Introductory Physics for the Health and Life Sciences II Copyright © 2012 by OSCRiceUniversity is licensed under a Creative Commons Attribution 4.0 International License, except where otherwise noted.

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