Explanation Given: Potential at the surface of the sphere of radius R is given as: V o = k cos 3 θ Formula used: Potential inside the sphere can be given as: V ( r , θ ) = ∑ l = 0 ∞ A l r l P l cos θ Potential outside the sphere can be given as V ( r , θ ) = ∑ l = 0 ∞ B l r l + 1 P l cos θ Calculation: The potential at the surface of the sphere V o = k cos 3 θ Where, k is the constant. V o = k ( 4 cos 3 θ − 3 cos θ ) The general expression for the Legendre polynomial is P n ( x ) = d n d x n ( x 2 − 1 ) n / 2 For P 1 ( cos θ ) P 1 ( cos θ ) = d 1 d x 1 ( cos 2 θ − 1 ) 1 / 2 P 1 ( cos θ ) = d d x sin θ P 1 ( cos θ ) = cos θ For P 3 ( cos θ ) P 3 ( cos θ ) = d 3 d x 3 ( cos 2 θ − 1 ) 3 / 2 P 3 ( cos θ ) = 1 2 ( 5 cos 3 θ − 3 cos θ ) In terms of Legendre polynomial, the potential of the surface of the sphere is V o = k ( α P 3 ( cos θ ) + β P 1 ( cos θ ) ) .... ( 1 ) α and β are the Legendre polynomial constants. Plugin the value of P 1 cos θ and P 3 cos θ k ( 4 cos 3 θ − 3 cos θ ) = k ( α [ 1 2 ( 5 cos 3 θ − 3 cos θ ) ] + β cos θ ) ( 4 cos 3 θ − 3 cos θ ) = ( α [ 1 2 ( 5 cos 3 θ − 3 cos θ ) ] + β cos θ ) ( 4 cos 3 θ − 3 cos θ ) = 5 α 2 cos 3 θ + ( β − 3 α 2 ) cos θ On comparing the above equation, 5 α 2 = 4 α = 8 5 And ( β − 3 α 2 ) = − 3 β − 3 2 ( 8 5 ) = − 3 β = − 3 + 12 5 β = − 3 5 From equation (1) V o = k ( 8 5 P 3 ( cos θ ) − 3 5 P 1 ( cos θ ) ) Potential inside the sphere ( r ≤ R ) can be given as: V ( r , θ ) = ∑ l = 0 ∞ A l r l P l cos θ Multiply both sides by P l ( cos θ ) ( sin θ ) P l ( cos θ ) ( sin θ ) V ( r , θ ) = ∑ l = 0 ∞ A l r l P l cos θ P l ( cos θ ) ( sin θ ) ∫ V ( r , θ ) P l ( cos θ ) ( sin θ ) d θ = A l r l ∫ ∑ l = 0 ∞ P l cos θ P l ( cos θ ) ( sin θ ) A l r l ( 2 l ' + 1 2 r ' ) = ∫ 0 π V ( r , θ ) P l ( cos θ ) ( sin θ ) d θ Using, ∫ ∑ l = 0 ∞ P l cos θ P l ( cos θ ) ( sin θ ) = { 2 2 l ' + 1 if l ' = 1 0 if l ' = 0 } In the above equation, A l r l = ( 2 l ' + 1 2 R ' ) ∫ 0 π V ( θ ) P l ( cos θ ) ( sin θ ) d θ A l r l = ( 2 l ' + 1 2 R ' ) [ ∫ 0 π k 5 { 8 P 3 ( cos θ ) − 3 P 1 ( cos θ ) } P l ( cos θ ) ( sin θ <

4th Edition

ISBN: 9781108420419

Author: David J. Griffiths

Publisher: Cambridge University Press

See similar textbooks

Concept explainers

Electric Charges and Fields

One of the fundamental properties is the electromagnetic property. Charge cannot be destroyed by any process and this contributes formally to the law of charge conservation.

Videos

Question

Chapter 3.3, Problem 3.19P

To determine

To Find: The potential inside and outside the sphere and surface charge density on the sphere.

Expert Solution & Answer

Want to see the full answer?

Check out a sample textbook solution

See solution

Chapter 3.3, Problem 3.18P

Chapter 3.3, Problem 3.20P

Students have asked these similar questions

Just 5 and 6 don't mind 7

In an electron gun, electrons are accelerated through a region with an electric field of magnitude 1.5 × 104 N/C for a distance of 2.5 cm. If the electrons start from rest, how fast are they moving after traversing the gun?

Please solve and answer this problem correctly please. Thank you!!

Chapter 3 Solutions

Introduction to Electrodynamics

Ch. 3.1 - Find the average potential over a spherical...Ch. 3.1 - Prob. 3.2P Ch. 3.1 - Prob. 3.3P Ch. 3.1 - Prob. 3.4P Ch. 3.1 - Prob. 3.5P Ch. 3.1 - Prob. 3.6P Ch. 3.2 - Find the force on the charge +q in Fig. 3.14....Ch. 3.2 - (a) Using the law of cosines, show that Eq. 3.17...Ch. 3.2 - In Ex. 3.2 we assumed that the conducting sphere...Ch. 3.2 - A uniform line charge is placed on an infinite...

Knowledge Booster

Learn more about

Need a deep-dive on the concept behind this application? Look no further. Learn more about this topic, physics and related others by exploring similar questions and additional content below.

Similar questions

SEE MORE QUESTIONS

Recommended textbooks for you

Physics for Scientists and Engineers: Foundations...
Physics
ISBN:9781133939146
Author:Katz, Debora M.
Publisher:Cengage Learning
Principles of Physics: A Calculus-Based Text
Physics
ISBN:9781133104261
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
University Physics Volume 2
Physics
ISBN:9781938168161
Author:OpenStax
Publisher:OpenStax
College Physics
Physics
ISBN:9781285737027
Author:Raymond A. Serway, Chris Vuille
Publisher:Cengage Learning
Physics for Scientists and Engineers, Technology ...
Physics
ISBN:9781305116399
Author:Raymond A. Serway, John W. Jewett
Publisher:Cengage Learning
Classical Dynamics of Particles and Systems
Physics
ISBN:9780534408961
Author:Stephen T. Thornton, Jerry B. Marion
Publisher:Cengage Learning