Charged Conducting Sphere in Constant Electric Field
المؤلف:
Sidney B. Cahn Boris E. Nadgorny
المصدر:
A GUIDE TO PHYSICS PROBLEMS
الجزء والصفحة:
part 1 , p 44
7-8-2016
1732
Charged Conducting Sphere in Constant Electric Field
A conducting sphere of radius on whose surface resides a total charge Q is placed in a uniform electric field E0 (see Figure 1.1). Find the potential
Figure 1.1
at all points in space exterior to the sphere. What is the surface charge density?
SOLUTION
Look for a solution of the form
where ϕ0 = -E0 · r is the potential due to the external field and ϕ1 is the change in the potential due to the presence of the sphere. The constant vector E0 defines a preferred direction, and therefore the potential ϕ1 may depend only on this vector. Then, the only solution of Laplace’s equation which goes to zero at infinity is a dipole potential
(1)
where A is some constant (alternatively, we may write the solution in terms of Legendre polynomials and obtain the same answer from the boundary conditions). So
(2)
On the surface of the sphere, ϕ is constant:
(3)
where θ is the angle between E0 and r (see Figure 1.2). From (3), we find that A = a3, and finally
(4)
The surface charge density
Figure 1.2
الاكثر قراءة في مواضيع اخرى
اخر الاخبار
اخبار العتبة العباسية المقدسة
