Problem 2: A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder? a) Compute the surface charge density n of the shell from its total charge and geometrical parameters. Vcenter = 1 Q 2 In 4л€ L t₂ S² b) Which charge dq is enclosed in a thin ring of width dz located at a distance z from the center of the cylinder (shown in Fig.2)? Which potential dV does this ring create at the center (you need to use the formula derived in the textbook for the potential of a charged ring along its axis). dz c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z. Show that (The integral that you need to use here is d dt √²+a² R² + 1/2 + 1/1/20 √√R² +4-4 L² R2 L R FIG. 2: The scheme for Problem 2 [2 = ln(t + √₁² + a²) 1².) 2
Problem 2: A hollow cylindrical shell of length L and radius R has charge Q uniformly distributed along its length. What is the electric potential at the center of the cylinder? a) Compute the surface charge density n of the shell from its total charge and geometrical parameters. Vcenter = 1 Q 2 In 4л€ L t₂ S² b) Which charge dq is enclosed in a thin ring of width dz located at a distance z from the center of the cylinder (shown in Fig.2)? Which potential dV does this ring create at the center (you need to use the formula derived in the textbook for the potential of a charged ring along its axis). dz c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z. Show that (The integral that you need to use here is d dt √²+a² R² + 1/2 + 1/1/20 √√R² +4-4 L² R2 L R FIG. 2: The scheme for Problem 2 [2 = ln(t + √₁² + a²) 1².) 2
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Transcribed Image Text:Problem 2: A hollow cylindrical shell of length L and radius R
has charge Q uniformly distributed along its length. What is the
electric potential at the center of the cylinder?
a) Compute the surface charge density n of the shell from its
total charge and geometrical parameters.
Vcenter =
1 Q
2 In
4л€ L
t₂
S²
b) Which charge dq is enclosed in a thin ring of width dz located at a distance z from the center of
the cylinder (shown in Fig.2)? Which potential dV does this ring create at the center (you need to use the
formula derived in the textbook for the potential of a charged ring along its axis).
dz
c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z.
Show that
(The integral that you need to use here is d
dt
√²+a²
R² + 1/2 + 1/1/20
√√R² +4-4
L²
R2
L
R
FIG. 2: The scheme for Problem 2
[2
= ln(t + √₁² + a²) 1².)
2
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