PHGN200: All Sections
Recitation 2
February 6, 2007
y
R
q
`
θ
x
`
−q
Figure 1: For the above electric dipole, p = 2`q.
1. Dipole in a 2-D world because the 3-D world is too damn hard!
(a) Find the electric potential anywhere in the xy-plane (see Fig. 1).
q
Ans: V = 4πo √ 2 1 2 − √ 2 1 2
x +(y−`)
x +(y+`)
(b) Find the x−component of the electric field from the expression you have obtained for the electric potential
in part (a). Is your result consistent with what you obtained during the first recitation?
h
i
x
x
−
Ans: Ex = 4πq o [x2 +(y−`)
3/2
3/2
2]
[x2 +(y+`)2 ]
2. Given a curve y = f (x) where x1 ≤ x ≤ x2 in the xy−plane, find the electric potential at some field point,
(a, b). Assume that the curve has a linear charge density given by λ = g(x).
q
R x2
λ
1
√
Ans: V = 4πo x1
1 + [f 0 (x)]2 dx, whereλ = g(x)
2
2
(a−x) +(b−f (x))
3. A sphere with radius R and volume charge density ρ = ρo rn , where ρo > 0 and n ≥ 0, is centered at the
origin of a coordinate system.
(a) Find the electric field inside the sphere, i.e., r < R.
Ans:
E=
ρo rn+1
,
(n+3)o
r < R.
(b) Find the electric field outside the sphere, i.e., r > R.
Ans:
E=
ρo Rn+3
,
r2 (n+3)o
r>R
(c) Find the electric potential difference between the center and surface of the sphere, i.e., Vcenter − Vsurface .
Ans:
Vcenter − Vsurface =
ρo Rn+2
(n+2)(n+3)o
4. A cylinder with radius R, length `, and volume charge density ρ = ρo rn , where ρo > 0 and n ≥ 0, is shown
in Fig. (2).
PHGN200: All Sections
Recitation 2
February 6, 2007
z
ℓ
z
θ
R
y
x
Figure 2: A cylinder with radius R, and length `, where
R
`
1. The figure is not to scale.
(a) Find the electric field inside the cylinder, i.e., r < R.
Ans:
E=
ρo rn+1
,
(n+2)o
r<R
(b) Find the electric field outside the cylinder, i.e., r > R.
Ans:
E=
ρo Rn+2
,
(n+2)ro
r>R
(c) Find the electric potential difference between the center and the surface of the cylinder, i.e., Vcenter −
Vsurface .
ρo Rn+2
Ans: Vcenter − Vsurface = (n+2)
2
o
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