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  5. Conductors in electrostatic

6.4 Conductors in electrostatic equilibrium  (Page 5/9)

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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 ( [link] ). 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 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.
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.

Summary

  • The electric field inside a conductor vanishes.
  • Any excess charge placed on a conductor resides entirely on the surface of the conductor.
  • The electric field is perpendicular to the surface of a conductor everywhere on that surface.
  • The magnitude of the electric field just above the surface of a conductor is given by E = σ ε 0 .

Key equations

Definition of electric flux, for uniform electric field Φ = E → · A → → E A cos θ
Electric flux through an open surface Φ = ∫ S E → · n ^ d A = ∫ S E → · d A →
Electric flux through a closed surface Φ = ∮ S E → · n ^ d A = ∮ S E → · d A →
Gauss’s law Φ = ∮ S E → · n ^ d A = q enc ε 0
Gauss’s Law for systems with symmetry Φ = ∮ S E → · n ^ d A = E ∮ S d A = E A = q enc ε 0
The magnitude of the electric field just outside the surface of a conductor E = σ ε 0

Conceptual questions

Is the electric field inside a metal always zero?

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Under electrostatic conditions, the excess charge on a conductor resides on its surface. Does this mean that all the conduction electrons in a conductor are on the surface?

yes

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A charge q is placed in the cavity of a conductor as shown below. Will a charge outside the conductor experience an electric field due to the presence of q ?

Figure shows an egg shape with an oval cavity within it. The cavity is surrounded by a dotted line just outside it. This is labeled S. There is a positive charge labeled q within the cavity.
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The conductor in the preceding figure has an excess charge of – 5. 0 µ C . If a 2.0 - µ C point charge is placed in the cavity, what is the net charge on the surface of the cavity and on the outer surface of the conductor?

Since the electric field is zero inside a conductor, a charge of −2.0 μ C is induced on the inside surface of the cavity. This will put a charge of + 2.0 μ C on the outside surface leaving a net charge of −3.0 μ C on the surface.

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Problems

An uncharged conductor with an internal cavity is shown in the following figure. Use the closed surface S along with Gauss’ law to show that when a charge q is placed in the cavity a total charge – q is induced on the inner surface of the conductor. What is the charge on the outer surface of the conductor?

A metal sphere with a cavity is shown. It is labeled vector E equal to zero. There are plus signs surrounding it. There is a positive charge labeled plus q within the cavity. The cavity is surrounded by minus signs.
A charge inside a cavity of a metal. Charges at the outer surface do not depend on how the charges are distributed at the inner surface since E field inside the body of the metal is zero.
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An uncharged spherical conductor S of radius R has two spherical cavities A and B of radii a and b , respectively as shown below. Two point charges + q a and + q b are placed at the center of the two cavities by using non-conducting supports. In addition, a point charge + q 0 is placed outside at a distance r from the center of the sphere. (a) Draw approximate charge distributions in the metal although metal sphere has no net charge. (b) Draw electric field lines. Draw enough lines to represent all distinctly different places.

Figure shows a sphere with two cavities. A positive charge qa is in one cavity and a positive charge qb is in the other cavity. A positive charge q0 is outside the sphere at a distance r from its center.


Figure shows a sphere with two cavities. A positive charge qa is in one cavity and a positive charge qb is in the other cavity. A positive charge q0 is outside the sphere at a distance r from its center.

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Source:  OpenStax, University physics volume 2. OpenStax CNX. Oct 06, 2016 Download for free at http://cnx.org/content/col12074/1.3
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