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 = 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). 1 4πεο L c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z. Show that dt (The integral that you need to use here is ² √₁²+a² In = dz R² + ²/1/2 + 1/1/20 L² √√R² + 4/12 - 11/1 2 In(t + √t² + a²) R FIG. 2: The scheme for Problem 2 & V 2². )

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 = 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). 1 4πεο L c) Sum up the contributions from all the rings along the cylinder by integrating dV with respect to z. Show that dt (The integral that you need to use here is ² √₁²+a² In = dz R² + ²/1/2 + 1/1/20 L² √√R² + 4/12 - 11/1 2 In(t + √t² + a²) R FIG. 2: The scheme for Problem 2 & V 2². )

University Physics Volume 2

18th Edition

ISBN:9781938168161

Author:OpenStax

Publisher:OpenStax

Chapter6: Gauss's Law

Section: Chapter Questions

Problem 86AP: Two non-conducting spheres of radii R1 and R2 are uniformly charged with charge densities p1 and p2...

See similar textbooks

Similar questions

Please answer this, with complete answer An isolated solid spherical conductor of radius 5.00cm is surrounded by dry air. It is given a charge and acquires potential V, with the potential at infinity assumed to be zero. a. Calculate the maximum magnitude can have. b. Explain clearly and concisely why there is a maximum.
What is the potential a perpendicular distance of z0 from the center of an annulus (a disk with a hole cut out) with inside radius a, outside radius b, and with a uniform surface charge density σ?
A sphere of radius R centered at the origin carries charge density ρ(r, θ) = k*(R/r^ 2) (R − 2r) sin θ where k is a constant, and r, θ are the usual spherical coordinates. Use the multipole expansion to find the approximate potential for points on the z-axis far from the sphere (z >> R).
Given an electric potential of find its corresponding electric field vector. Sol. Using the concept of potential gradient, we have Since we only have, radial direction (r). Then, the del-operator will be By substitution, we get Evaluating the differential, we, get the following *(q/)
Fourpointcharges,eachwithchargeq=+2.0nC,arepositionedequidistantlyaroundacircle of radius r = 5.0 cm as shown in figure (A) below. (a) What is the electric field E at C? (b) What is the potential at point C located at the center (assuming V = 0 infintely far away)? If charge q2 is moved to the same position as q4, as shown in figure (B), what are the new (c) electric field E (d) andpotential at point C?
Find the potential inside and outside a uniformly charged solid spherewhose radius is R and whose total charge is q. Use infinity as your reference point.Compute the gradient of V in each region, and check that it yields the correct field.Sketch V (r).
A conducting sphere of radius a, at potential V0, is surrounded by a thin con- centric ring of radius b and charge Q , The Method of Images ،find potential point P.
Two non-conducting spheres of radii R1 and R2 are uniformly charged with charge densities p1 and p2 , respectively. They are separated at center-to-center distance a (see below). Find the electric field at point P located at a distance r from the center of sphere 1 and is in the direction from the line joining the two spheres assuming their charge densities are not affected by the presence of the other sphere. (Hint: Work one sphere at a time and use the superposition principle.)
A thin plastic rod of length L has a positive charge Q uniformly distributed along its length. We willcalculate the exact field due to the rod in the next homework set. In this set, we will approximatethe rod as several point sources and develop the Riemann sum as an intermediate step on the wayto writing an integral.For those aiming at a P rating, you may use L = 3.0m , Q = 17 mC, and y = 0.11m to calculate theanswer numerically first and substitute variables for them only as required in the problem statement.For those aiming at an E rating, leave L, Q and y as variables. Substitute numbers only whererequired in the problem statement, and only as a last step
Derive this last equation for disk of charge for Potential at point P.
Show that electric potential for a shell whose radius is R, has charge q uniformly distributed on its entire surface, is the same as electric potential for a conductor (Solid) has radius R and charge q.
Find the electric potential at a distance h from the center of the mantle of a cylinder of radius R, with a distribution of homogeneous charge σ. Discuss the cases when h < R and when h > R