Re: Charging by Induction

[John Mallinckrodt <ajmallinckro@CSUPOMONA.EDU>]

On Mon, 15 Feb 1999, Hasan Fakhruddin wrote:

> Say an object carrying a charge +q1 induces charge q2 on another object
> (-q2 on the nearer end and +q2 on the farther end).
>
> Comparing only the magnitudes, under what conditions is
>
> q2 = q1,
> q2 > q1, and
> q2 < q1  ?

There is likely no simple answer to this question, but a number of general
results follow easily from applying Gauss' Law which is simply a
formalized way of doing some field line accounting.

For simplicity's sake, let's assume that both of the objects are
conductors.  We then have the following results:

1. No field line can start and end on the same object. (They are
equipotentials.)

2. Far from the objects all field lines run outward.  (Net positive
charge.)

3. At least *some* field lines start on the object with charge +q1 (object
A) and end on the neutral object (object B).  (Assuming B polarizes at
all, the field lines that end on the negative charge can *only* start on
A.  They can't start on B and they can't come in from infinity.)

4. *No* field lines can end on object A.  (They'd have to come either from
infinity which is not possible or from object B which we already know is
at a lower potential from item 3.)

5. No more than *all* of the field lines "starting" on the object with
charge +q1 (object A) can "end" on the uncharged object (object B).

The best thing is to draw some pictures yourself.

Conclusions:  q2 is less than or equal to q1.  It will be equal only if
the geometry is such that B captures *all* of the field lines leaving A;
for instance, if A is inside a cavity in B.  In general q2 will get
smaller as B wraps less around and/or moves away from A.

For nonconductors the details get more difficult, but I would certainly
expect q2 to be less than it would be for an equivalent geometry formed
from conductors.

John
-----------------------------------------------------------------
 A. John Mallinckrodt      http://www.csupomona.edu/~ajm
 Professor of Physics    mailto:ajm@csupomona.edu
 Physics Department       voice:909-869-4054
 Cal Poly Pomona            fax:909-869-5090
 Pomona, CA 91768-4031   office:Building 8, Room 223