Solving 04 PDF

Preview

Summary

Problem Solving...

Description

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02

Problem Solving 4: Conductors, Capacitance, and Stored Energy Section ______Table ________________________ Names ____________________________________ ____________________________________ ____________________________________ ____________________________________

Hand in one copy per group at the end of the Friday Problem Solving Session. OBJECTIVES 1. To calculate the capacitance for symmetric configurations of conductors. 2. To calculate the energy stored in a capacitor. REFERENCE: Course Notes: Sections 5.1-5.6, 5.8-5.9

PROBLEM SOLVING STRATEGIES (see Section 5.8, 8.02 Course Notes) (1) Using Gauss’s Law, calculate the electric field everywhere. (2) Compute the electric potential difference ΔV between the two conductors. (3) Calculate the capacitance C using C = Q / | ΔV | . (4) Calculate the stored energy by integrating the energy density over the regions of space with non-zero energy density,

U E = ∫ uE dV =

1 ε0 E 2 dV . ∫ 2

(5) Equivalently, calculate the stored energy by U E = (1/ 2)C (ΔV )2 or U E = Q 2 / 2C .

Solving4-1

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02 Problem 1: Faraday Ice Pail Consider two nested cylindrical conductors of height h and radii a & b respectively. A charge +Q is evenly distributed on the outer surface of the pail (the inner cylinder), -Q on the inner surface of the shield (the outer cylinder).

(a) Calculate the electric field between the two cylinders (a < r < b). Show all your work.

Solving4-2

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02 (b) Calculate the potential difference between the two cylinders. Show all your work.

(c) Calculate the capacitance of this system, using C = Q / ΔV .

(d) Numerically evaluate the capacitance for the experimental setup, given: h ≅ 15 cm, a ≅ 4.75 cm and b ≅ 7.25 cm.

Solving4-3

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02 (e) Find the electric field energy density at any point between the conducting cylinders. How much energy resides in a cylindrical shell between the conductors of radius r (with a < r < b ), height h , thickness dr , and volume 2π rh dr ? Integrate your expression to find the total energy stored in the capacitor and compare your result with that obtained using U E = (1/ 2)C (ΔV )2 .

Solving4-4

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02 Problem 2: Spherical Shells: Capacitance and Maximum Charge Consider a spherical vacuum capacitor consisting of inner and outer thin conducting spherical shells with charge +Q on the inner shell of radius a = 0.10 m and charge −Q on the outer shell of radius b = 0.20 m . You may neglect the thickness of each shell. You may use the result that 1/(4πε 0 ) = 9.0 × 109 N ⋅ m 2 ⋅C−2 . In this problem first express your answers in terms of r , ε 0 , Q , a , and b as necessary, and then substitute in values for the given quantities.

spherical shell with charge Q

b a spherical shell with charge + Q

a) Determine an expression for the magnitude of the electric field in each of the regions i) r < a , (ii) a < r < b , and (iii) r > b . Clearly indicate the direction of the electric field on the above figure. i.

ii.

For r < a :

For a < r < b :

Solving4-5

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02

iii.

For r > b :

b) What is the electric potential difference V (a) − V (b) ? Do you expect this potential difference to be positive or negative?

c) What is the capacitance C of this object?

Solving4-6

MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics: 8.02

d) Show that in the limit when the distance between the shells is very small lim (b − a) → δ...