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Date: 30-8-2016
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Potential of Charged Rod
A thin non-conducting rod of length L carries a uniformly distributed charge Q and is oriented as shown in Figure 1.1.
Figure 1.1
a) Find the potential ϕ due to the charged rod for any point on the x-axis with z > L/2.
b) Find ϕ(r, θ, φ) for all |r| > L/2 where r, θ, φ are the usual spherical coordinates.
Hint: The general solution to Laplace’s equation in spherical coordinates is
(i)
SOLUTION
a) The potential along the z-axis may be computed by integrating along the rod:
(1)
where u ≡ L/2z.
b) Since the problem has azimuthal symmetry, we may use the Legendre polynomials Pl (cosθ) and rewrite the expansion in (i) (see Figure 1.2)
Since we consider r > L/2, all the Al are zero, and
(2)
Figure 1.2
We wish to equate (2) along the with z-axis (1) where Pl(1) = 1 for all l. We must rewrite ln|(1+u)/(1-u)| as a sum. Now,
So
(3)
Rewriting (1) using (3), we find
Replacing n by l + 1,
(4)
Using (2) and (4), we have
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مخاطر عدم علاج ارتفاع ضغط الدم
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اختراق جديد في علاج سرطان البروستات العدواني
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مدرسة دار العلم.. صرح علميّ متميز في كربلاء لنشر علوم أهل البيت (عليهم السلام)
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