Department of Energy Argonne National Laboratory Office of Science NEWTON's Homepage NEWTON's Homepage
NEWTON, Ask A Scientist!
NEWTON Home Page NEWTON Teachers Visit Our Archives Ask A Question How To Ask A Question Question of the Week Our Expert Scientists Volunteer at NEWTON! Frequently Asked Questions Referencing NEWTON About NEWTON About Ask A Scientist Education At Argonne Hollow Conductor Field
Name: Paul
Status: student
Age: 16
Location: N/A
Country: N/A
Date: 10/2/2004


Question:
What I do not understand is why the intensity of the electric field is 0 inside a charged (empty on the inside) metal sphere. I know that the electrical charge is spread uniformly across its outer surface ( what exactly does that have to do with it?) and that anywhere outside E=k*q*q/(r*r), but why, for example,does E(a)=E(b)=0? (A and B are two points from inside the sphere) Shouldn't the local electric charge closest to A create a very high intensity field (~/(r^2); r->0 => E->oo) in A?; while in point B (I considered point B in the middle of the sphere), because of the symmetry, it is clear that E=0, thus E(a)>E(b)=0...?? I am confused; why does Coulomb's Law not apply here as well? If we were to consider, let us say, 50 electrified particles, placed in the form of a sphere, would it be the same as if they were on an empty metal sphere with the same radius? I guess not, but why?

I know it has to do with the sphere being a metal conductor (electrons moving freely inside and "stuff like that"), but my teacher simply says: "That is how things are." So how can I understand? I know "that's how things are", but why?


Replies:
Paul,

Try to picture a point near the top of a charged hollow sphere, on the inside. It is true that the charges above the location are closer. Each charge exerts a large force. What you may not realize is that there is a many more charges below the location. Although these charges are further away, there are many more charges trying to make the electric field at least partially upward. Greater charge below balances greater force per charge above.

Ken Mellendorf
Math, Science, Engineering
Illinois Central College


It is zero because

1) The field goes as 1/r^2.

2) The surface charge on the sphere's surface is uniform, thus charge is proportional to surface area.

3) The charged regions that oppose each other (i.e., those on opposite sides of the spherical surface from any given point inside) occupy areas that increase as the square of the distance from the given point.

Here is a mental picture for the underlying calculus:

Draw a very small circle on the surface of the sphere, and pick any point inside the sphere. Call the point "P". Let's say the point P is at distance 'd' from the circle you drew. Now draw lines from every point on the circle through the point P, and see where those lines intersect the opposite side of the sphere's surface. The intersection will also be a very small circle, at a distance we'll call 'D'. (If the intersection on the other side is not a very small circle, you need to make the first circle smaller. You're going to have to get infinitesimal.)

If the circle you drew is small enough, its area will be proportional to d^2 (i.e., we will not have to worry about the fact that the sphere's surface is curved), and the area of the circle on the other side will be proportional to D^2.

Now, since charge is uniformly distributed over the spherical surface, the total charges within the two circles are proportional to their areas. So the field from the first area is proportional to charge/d^2 ~ area/d^2 ~ d^d/d^2 = 1. (I'm using '~' to mean 'proportional to'.)

Similarly, the opposing field from the other side of the sphere is proportional to D^2/D^2 = 1, so they balance each other exactly.

Note that, as the point A approaches the surface, and 1/d^2 approaches infinity, the charge within the circle at distance d goes to zero. So the field contribution from the near-side circle remains finite, and it continues to be balanced by the field contribution from the far-side circle.

Tim Mooney


Things are the way they are for very good reasons. For example, the electric field due to a charge falls off by an inverse square law, as you state in your email. It is that way because we live in a three dimensional universe and the area of a sphere in our universe increases proportionally to the square of its radius (Area = 4*pi*R^2). So a charge at the center of a mathematical sphere (no other charges and no conductors or other material nearby) produces a field at the surface of the sphere over an area that grows like the square of the distance from the charge. Therefore, the effect per unit area has to decrease like the inverse square of the radius. The same is true of gravity.

If you draw straight lines, say every 30 degrees, coming from a point charge and draw circles of various radii centered on that point, you can probably convince yourself that in a two-dimensional universe (flatland) the electric field due to a point charge would fall off like 1/R,

As for the field inside a charged conducting sphere, your argument that the field must be zero at the center because of the symmetry is exactly right (and shows good understanding). When you consider a point off center, however, the field is still zero. This is because although the charge on the near side of the sphere is closer, less of it is effective. To see this, consider a cone with the apex at the off center point. For each bit of charge inside the cone on the near side, there is a larger amount of charge inside the cone on the far side. And since the cross-sectional area of a cone grows like the square of the distance from the apex, the relative amounts of charge exactly balances the relative distances. If you sketch a sphere (start with a circle) and draw some two ended cones, it will be a lot clearer.

Fifty charges uniformly spaced over the surface of a mathematical sphere would have almost the same effects, except that the gaps between the charges will have some minor effects. If you had a million charges, it would act almost identically. Remember that a charged conducting sphere probably has more than 1.0E15 electrons uniformly spread over its surface (that is 1 followed by 15 zeros).

Best, Dick Plano, Professor of Physics emeritus, Rutgers University



Click here to return to the Physics Archives

NEWTON is an electronic community for Science, Math, and Computer Science K-12 Educators, sponsored and operated by Argonne National Laboratory's Educational Programs, Andrew Skipor, Ph.D., Head of Educational Programs.

For assistance with NEWTON contact a System Operator (help@newton.dep.anl.gov), or at Argonne's Educational Programs

NEWTON AND ASK A SCIENTIST
Educational Programs
Building 360
9700 S. Cass Ave.
Argonne, Illinois
60439-4845, USA
Update: June 2012
Weclome To Newton

Argonne National Laboratory