2.4 Conductors in Electrostatic Equilibrium

Samuel J. Ling; Jeff Sanny; William Moebs; Daryl Janzen

2.4 Conductors in Electrostatic Equilibrium

LEARNING OBJECTIVES

By the end of this section, you will be able to:

Describe the electric field within a conductor at equilibrium
Describe the electric field immediately outside the surface of a charged conductor at equilibrium
Explain why if the field is not as described in the first two objectives, the conductor is not at equilibrium

So far, we have generally been working with charges occupying a volume within an insulator. We now study what happens when free charges are placed on a conductor. Generally, in the presence of a (generally external) electric field, the free charge in a conductor redistributes and very quickly reaches electrostatic equilibrium. The resulting charge distribution and its electric field have many interesting properties, which we can investigate with the help of Gauss’s law and the concept of electric potential.

The Electric Field inside a Conductor Vanishes

If an electric field is present inside a conductor, it exerts forces on the free electrons (also called conduction electrons), which are electrons in the material that are not bound to an atom. These free electrons then accelerate. However, moving charges by definition means nonstatic conditions, contrary to our assumption. Therefore, when electrostatic equilibrium is reached, the charge is distributed in such a way that the electric field inside the conductor vanishes.

If you place a piece of a metal near a positive charge, the free electrons in the metal are attracted to the external positive charge and migrate freely toward that region. The region the electrons move to then has an excess of electrons over the protons in the atoms and the region from where the electrons have migrated has more protons than electrons. Consequently, the metal develops a negative region near the charge and a positive region at the far end (Figure 2.4.1). As we saw in the preceding chapter, this separation of equal magnitude and opposite type of electric charge is called polarization. If you remove the external charge, the electrons migrate back and neutralize the positive region.

(Figure 2.4.1) $\begin{gather*}.\end{gather*}$

Figure shows a sphere and a positive charge q some distance away from it. The side of the sphere facing q is labeled A and the opposite side is labeled B. Minus signs and plus signs are shown at the inner surfaces of the sphere on sides A and B respectively. These are labeled minus sigma A and plus sigma B respectively. — **Figure 2.4.1** Polarization of a metallic sphere by an external point charge $+q$ . The near side of the metal has an opposite surface charge compared to the far side of the metal. The sphere is said to be polarized. When you remove the external charge, the polarization of the metal also disappears.

The polarization of the metal happens only in the presence of external charges. You can think of this in terms of electric fields. The external charge creates an external electric field. When the metal is placed in the region of this electric field, the electrons and protons of the metal experience electric forces due to this external electric field, but only the conduction electrons are free to move in the metal over macroscopic distances. The movement of the conduction electrons leads to the polarization, which creates an induced electric field in addition to the external electric field (Figure 2.4.2). The net electric field is a vector sum of the fields of $+q$ and the surface charge densities $-\sigma_A$ and $+\sigma_B$ This means that the net field inside the conductor is different from the field outside the conductor.

(Figure 2.4.2) $\begin{gather*}.\end{gather*}$

Figure shows a sphere and a charge q some distance away from it. The side of the sphere facing q is labeled A and the opposite side is labeled B. The inner surfaces of the sphere on sides A and B are labeled minus sigma A and plus sigma B respectively. A point P is on the sphere. Two arrows originate from P. They are labeled vector E subscript A and vector E subscript B. A dotted line bisects the angle formed by the two and connects P to q. A third arrow originates from P and points in the direction opposite to q. This is labeled vector E subscript q. — **Figure 2.4.2** In the presence of an external charge $q$ , the charges in a metal redistribute. The electric field at any point has three contributions, from $+q$ and the induced charges $-\sigma_A$ and $+\sigma_B$ . Note that the surface charge distribution will not be uniform in this case.

The redistribution of charges is such that the sum of the three contributions at any point $P$ inside the conductor is

$\vec{\mathbf{E}}_P=\vec{\mathbf{E}}_q+\vec{\mathbf{E}}_B+\vec{\mathbf{E}}_A=\vec{\mathbf{0}}.$

Now, thanks to Gauss’s law, we know that there is no net charge enclosed by a Gaussian surface that is solely within the volume of the conductor at equilibrium. That is, $q_{\mathrm{enc}}=0$ and hence

(2.4.1) $\begin{equation*}\vec{\mathbf{E}}_{\mathrm{net}}=\vec{\mathbf{0}}\ (\mathrm{at~points~inside~a~conductor}).\end{equation*}$

Charge on a Conductor

An interesting property of a conductor in static equilibrium is that extra charges on the conductor end up on the outer surface of the conductor, regardless of where they originate. Figure 2.4.3 illustrates a system in which we bring an external positive charge inside the cavity of a metal and then touch it to the inside surface. Initially, the inside surface of the cavity is negatively charged and the outside surface of the conductor is positively charged. When we touch the inside surface of the cavity, the induced charge is neutralized, leaving the outside surface and the whole metal charged with a net positive charge.

(Figure 2.4.3) $\begin{gather*}.\end{gather*}$

A figure on the left shows a shaded circle with a cavity in it. A rod with a ball at the end is inserted in the cavity in such a way that it does not touch the shaded circle. The ball has a plus sign on it. The cavity has minus signs around it. The shaded circle has plus signs outside it. An arrow points from this figure to a figure on the right. The arrow is labeled touch inside cavity. The figure on the right is similar to the figure on the left, except that the ball is touching the edge of the cavity. There are no signs on the ball or around the cavity. The outside of the shaded circle has plus signs. — Figure 2.4.3 Electric charges on a conductor migrate to the outside surface no matter where you put them initially.

To see why this happens, note that the Gaussian surface in Figure 2.4.4 (the dashed line) follows the contour of the actual surface of the conductor and is located an infinitesimal distance within it. Since $E=0$ everywhere inside a conductor,

$\oing_S\vec{\mathbf{E}}\cdot\hat{\mathbf{n}}dA=0.$

Thus, from Gauss’ law, there is no net charge inside the Gaussian surface. But the Gaussian surface lies just below the actual surface of the conductor; consequently, there is no net charge inside the conductor. Any excess charge must lie on its surface.

(Figure 2.4.4) $\begin{gather*}.\end{gather*}$

Figure shows an irregular shape. A dotted line is shown just inside the outline of the shape. — **Figure 2.4.4** The dashed line represents a Gaussian surface that is just beneath the actual surface of the conductor.

This particular property of conductors is the basis for an extremely accurate method developed by Plimpton and Lawton in 1936 to verify Gauss’s law and, correspondingly, Coulomb’s law. A sketch of their apparatus is shown in Figure 2.4.5. Two spherical shells are connected to one another through an electrometer $\mathrm{E}$ , a device that can detect a very slight amount of charge flowing from one shell to the other. When switch $\mathrm{S}$ is thrown to the left, charge is placed on the outer shell by the battery $\mathrm{B}$ . Will charge flow through the electrometer to the inner shell?

No. Doing so would mean a violation of Gauss’s law. Plimpton and Lawton (1936) did not detect any flow and, knowing the sensitivity of their electrometer, concluded that if the radial dependence in Coulomb’s law were $1/r^{2+\delta}$ , $\delta$ would be less than $2\times10^{-9}$ . More recent measurements by Williams, Faller, and Hill (1971) place $\delta$ at less than $3\times10^{-16}$ , a number so small that the validity of Coulomb’s law seems indisputable.

(Figure 2.4.5) $\begin{gather*}.\end{gather*}$

Figure shows a circle labeled E. It is surrounded by two concentric circles with slits in them. These are labeled two concentric conducting spheres. Two terminals of E are connected, one to each circle. The outer circle is connected to a switch S, which switches between two terminals of a battery. There is a slanted plate at the top of the circles labeled mirror for viewing the electrometer. A person views the mirror through a scope. The line of view is reflected from the mirror to E. — **Figure 2.4.5** A representation of the apparatus used by Plimpton and Lawton. Any transfer of charge between the spheres is detected by the electrometer $\mathrm{E}$ .

The Electric Field at the Surface of a Conductor

If the electric field had a component parallel to the surface of a conductor, free charges on the surface would move, a situation contrary to the assumption of electrostatic equilibrium. Therefore, the electric field is always perpendicular to the surface of a conductor.

At any point just above the surface of a conductor, the surface charge density $\sigma$ and the magnitude of the electric field $E$ are related by

(2.4.2) $\begin{equation*}E=\frac{\sigma}{\epsilon_0}.\end{equation*}$

To see this, consider an infinitesimally small Gaussian cylinder that surrounds a point on the surface of the conductor, as in Figure 2.4.6. The cylinder has one end face inside and one end face outside the surface. The height and cross-sectional area of the cylinder are $\delta$ and $\Delta A$ , respectively. The cylinder’s sides are perpendicular to the surface of the conductor, and its end faces are parallel to the surface. Because the cylinder is infinitesimally small, the charge density $\sigma$ is essentially constant over the surface enclosed, so the total charge inside the Gaussian cylinder is $\sigma\Delta A$ . Now $E$ is perpendicular to the surface of the conductor outside the conductor and vanishes within it, because otherwise, the charges would accelerate, and we would not be in equilibrium. Electric flux therefore crosses only the outer end face of the Gaussian surface and may be written as $E\Delta A$ , since the cylinder is assumed to be small enough that $E$ is approximately constant over that area. From Gauss’ law,

$E\Delta A=\frac{\sigma\Delta A}{\epsilon_0}.$

Thus,

$E=\frac{\sigma}{\epsilon_0}.$

(Figure 2.4.6) $\begin{gather*}.\end{gather*}$

A surface labeled sigma has plus signs on it. A point P on the surface forms the center of a cylinder. An arrow labeled vector E is along the axis of the cylinder and emerges from its top surface. The top surface of the cylinder is labeled delta A and the bottom surface is labeled vector E equal to zero. These are parallel to the surface sigma. The length of the cylinder is labeled sigma. — **Figure 2.4.6** An infinitesimally small cylindrical Gaussian surface surrounds point $P$ , which is on the surface of the conductor. The field $\vec{\mathbf{E}}$ is perpendicular to the surface of the conductor outside the conductor and vanishes within it.

EXAMPLE 2.4.1

Electric Field of a Conducting Plate

The infinite conducting plate in Figure 2.4.7 has a uniform surface charge density $\sigma$ . Use Gauss’ law to find the electric field outside the plate. Compare this result with that previously calculated directly.

(Figure 2.4.7) $\begin{gather*}.\end{gather*}$

A shaded strip labeled E equal to zero has plus signs on both its inner edges. A rectangle labeled A is shown on the right of the strip such that it encloses two plus signs. Two arrows within this are perpendicular to the length of the strip and point right. These are labeled vector E. — **Figure 2.4.7** A side view of an infinite conducting plate and Gaussian cylinder with cross-sectional area $A$ .

Strategy

For this case, we use a cylindrical Gaussian surface, a side view of which is shown.

Solution

The flux calculation is similar to that for an infinite sheet of charge from the previous chapter with one major exception: The left face of the Gaussian surface is inside the conductor where $\vec{\mathbf{E}}=\vec{\mathbf{0}}$ , so the total flux through the Gaussian surface is $EA$ rather than $2EA$ . Then from Gauss’ law,

$EA=\frac{\sigma A}{\epsilon_0}$

and the electric field outside the plate is

$E=\frac{\sigma}{\epsilon_0}.$

Significance

This result is in agreement with the result from the previous section, and consistent with the rule stated above.

EXAMPLE 2.4.2

Electric Field between Oppositely Charged Parallel Plates

Two large conducting plates carry equal and opposite charges, with a surface charge density $\sigma$ of magnitude $6.81\times10^{-7}~\mathrm{C/m}^2$ , as shown in Figure 2.4.8. The separation between the plates is $l=6.50~\mathrm{mm}$ . What is the electric field between the plates?

(Figure 2.4.8) $\begin{gather*}.\end{gather*}$

Two parallel plates are shown, a distance l apart. The left one has plus signs on the inside of its right surface. the right plate has minus signs on the inside of its left surface. Arrows from the left plate to the right plate are labeled vector E. A positive charge between the plates has an arrow from it, pointing right. — **Figure 2.4.8** The electric field between oppositely charged parallel plates. A test charge is released at the positive plate.

Strategy

Note that the electric field at the surface of one plate only depends on the charge on that plate. Thus, apply $E=\sigma/\epsilon_0$ with the given values.

Solution

The electric field is directed from the positive to the negative plate, as shown in the figure, and its magnitude is given by

$E=\frac{\sigma}{\epsilon_0}=\frac{6.81\times10^{-7}~\mathrm{C/m}^2}{8.85\times10^{-12}~\mathrm{C}^2/\mathrm{N~m}^2}=7.69\times10^4~\mathrm{N/C}.$

Significance

This formula is applicable to more than just a plate. Furthermore, two-plate systems will be important later.

EXAMPLE 2.4.3

A Conducting Sphere

The isolated conducting sphere (Figure 2.4.9) has a radius $R$ and an excess charge $q$ . What is the electric field both inside and outside the sphere?

(Figure 2.4.9) $\begin{gather*}.\end{gather*}$

Two concentric circles are shown. The smaller one, with radius R, has plus signs around the inside of it. The bigger one, with radius r is shown with a dotted line and labeled S, Gaussian surface. — **Figure 2.4.9** An isolated conducting sphere.

Strategy

The sphere is isolated, so its surface change distribution and the electric field of that distribution are spherically symmetrical. We can therefore represent the field as $\vec{\mathbf{E}}=E(r)\hat{\mathbf{r}}$ . To calculate $E(r)$ , we apply Gauss’s law over a closed spherical surface $S$ of radius $r$ that is concentric with the conducting sphere.

Solution

Since $r$ is constant and $\hat{\mathbf{n}}=\hat{\mathbf{r}}$ on the sphere,

$\oint_S\vec{\mathbf{E}}\cdot\hat{\mathbf{n}}dA=E(r)\oint_SdA=E(r)4\pi r^2.$

r<R

$E(r)=0,$

as expected inside a conductor. If $r>R$ , $S$ encloses the conductor so $q_{\mathrm{enc}}$ . From Gauss’s law,

$E(r)4\pi r^2=\frac{q}{\epsilon_0}.$

The electric field of the sphere may therefore be written as

$\begin{array}{ll}\vec{\mathbf{E}}=\vec{\mathbf{0}}&(r<R),\\\vec{\mathbf{E}}=\frac{1}{4\pi\epsilon_0}\frac{q}{r^2}\hat{\mathbf{r}}&(r\geq R).\end{array}$

Significance

Notice that in the region $r\geq0$ , the electric field due to a charge $q$ placed on an isolated conducting sphere of radius $R$ is identical to the electric field of a point charge $q$ located at the centre of the sphere. The difference between the charged metal and a point charge occurs only at the space points inside the conductor. For a point charge placed at the centre of the sphere, the electric field is not zero at points of space occupied by the sphere, but a conductor with the same amount of charge has a zero electric field at those points (Figure 2.4.10). However, there is no distinction at the outside points in space where $r>R$ , and we can replace the isolated charged spherical conductor by a point charge at its centre with impunity.

(Figure 2.4.10) $\begin{gather*}.\end{gather*}$

A circle labeled vector E subscript in equal to zero is shown. Arrows around it radiate outwards. These are labeled vector E subscript out. — **Figure 2.4.10** Electric field of a positively charged metal sphere. The electric field inside is zero, and the electric field outside is same as the electric field of a point charge at the center, although the charge on the metal sphere is at the surface.

CHECK YOUR UNDERSTANDING 2.6

How will the system above change if there are charged objects external to the sphere?

For a conductor with a cavity, if we put a charge $+q$ inside the cavity, then the charge separation takes place in the conductor, with $-q$ amount of charge on the inside surface and a $+q$ amount of charge at the outside surface (Figure 2.4.11(a)). For the same conductor with a charge $+q$ outside it, there is no excess charge on the inside surface; both the positive and negative induced charges reside on the outside surface (Figure 2.4.11(b)).

(Figure 2.4.11) $\begin{gather*}.\end{gather*}$

Figure a shows a metal sphere with a cavity within it. The sphere is labeled vector E equal to zero. It has plus signs around it. The cavity has minus signs around it. A positive charge plus q is within the cavity. Figure b shows the same metal sphere with a cavity in it. The sphere is labeled vector E equal to zero. There is nothing within the cavity. A positive charge labeled plus q is outside the sphere. The side of the sphere facing q has minus signs on it. The opposite side has plus signs on it. — **Figure 2.4.11** (a) A charge inside a cavity in a metal. The distribution of charges at the outer surface does not depend on how the charges are distributed at the inner surface, since the $E$ -field inside the body of the metal is zero. That magnitude of the charge on the outer surface does depend on the magnitude of the charge inside, however. (b) A charge outside a conductor containing an inner cavity. The cavity remains free of charge. The polarization of charges on the conductor happens at the surface.

If a conductor has two cavities, one of them having a charge $+q_a$ inside it and the other a charge $-q_b$ , the polarization of the conductor results in $-q_a$ on the inside surface of the cavity $a$ , $+q_b$ on the inside surface of the cavity $b$ , and $q_a-q_b$ on the outside surface (Figure 2.4.12). The charges on the surfaces may not be uniformly spread out; their spread depends upon the geometry. The only rule obeyed is that when the equilibrium has been reached, the charge distribution in a conductor is such that the electric field by the charge distribution in the conductor cancels the electric field of the external charges at all space points inside the body of the conductor.

(Figure 2.4.12) $\begin{gather*}.\end{gather*}$

Figure shows a flattened sphere, labeled vector E equal to zero. It has two spherical cavities within it. Its outer surface of the flattened sphere is labeled no induced charge outside. The left cavity has a negative charge q inside it, on the left. The left surface of this cavity has many plus signs on it and the right surface has a single plus sign on it. The right cavity has a positive charge q inside it, on the right. The right surface of this cavity has many minus signs on it and the left surface has a single minus sign on it. — **Figure 2.4.12** The charges induced by two equal and opposite charges in two separate cavities of a conductor. If the net charge on the cavity is nonzero, the external surface becomes charged to the amount of the net charge.

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