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A conducting sphere

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

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.
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 E = E ( r ) 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 n ^ = r ^ on the sphere,

S E · n ^ d A = E ( r ) S d A = E ( r ) 4 π r 2 .

For r < R , S is within the conductor, so q enc = 0 , and Gauss’s law gives

E ( r ) = 0 ,

as expected inside a conductor. If r > R , S encloses the conductor so q enc = q . From Gauss’s law,

E ( r ) 4 π r 2 = q ε 0 .

The electric field of the sphere may therefore be written as

E = 0 ( r < R ) , E = 1 4 π ε 0 q r 2 r ^ ( r R ) .

Significance

Notice that in the region r R , 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 center 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 center 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 ( [link] ). 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 center with impunity.

A circle labeled vector E subscript in equal to zero is shown. Arrows around it radiate outwards. These are labeled vector E subscript out.
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.
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Check Your Understanding How will the system above change if there are charged objects external to the sphere?

If there are other charged objects around, then the charges on the surface of the sphere will not necessarily be spherically symmetrical; there will be more in certain direction than in other directions.

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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 ( [link] (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 ( [link] (b)).

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.
(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.

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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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