What does Griffith mean by this hint? [Electrodynamics]

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SUMMARY

The discussion revolves around a question from Griffith's 'Introduction to Electrodynamics' regarding the implications of variable conductivity (sigma) on current density and charge density in a resistive medium. The participant acknowledges that the divergence of current density is non-zero due to the position-dependent conductivity. They seek clarification on the meaning of non-zero charge density in this context and are advised to utilize the constitutive equation j = σE and the continuity equation ∇ · j = 0 to modify Equation 7.5. Additionally, there is a suggestion to clarify the variable 's' as the radial coordinate in cylindrical coordinates.

PREREQUISITES
  • Understanding of Griffith's 'Introduction to Electrodynamics'
  • Familiarity with the concepts of current density and charge density
  • Knowledge of constitutive equations in electromagnetism
  • Basic grasp of continuity equations for stationary currents
NEXT STEPS
  • Study the implications of variable conductivity in electromagnetic fields
  • Learn how to apply the constitutive equation j = σE in different contexts
  • Explore the continuity equation ∇ · j = 0 and its applications in stationary currents
  • Review cylindrical coordinates and their relevance in electromagnetism
USEFUL FOR

This discussion is beneficial for physics students, particularly those studying electromagnetism, as well as educators and anyone seeking to deepen their understanding of current density and charge density in variable conductivity scenarios.

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I was solving the problems given by Griffith in his book 'Introduction to Electrodynamics' and stumbled across this question.
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"Because sigma (conductivity of the medium) is a function of position, the equation 7.5 does not hold" --i get this point; current density isn't constant and so its divergence will be non zero.
"The charge density is non zero in the resistive medium"-- I have no clue what this sentence means.. I can't even think why i don't find it right.. someone please explain it to me!

Thanks a lot !
 
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Another hint: Use the constitutive equation
$$\vec{j}=\sigma \vec{E}$$
and the continuity equation for stationary currents
$$\vec{\nabla} \cdot \vec{j}=0,$$
which together immediately tell you, how to modify Eq. (7.5). Also it would be helpful to tell, what ##s## is. Is it the cylinder coordinate ##r## in standard notation? I guess so from the Figure, which indicates that ##\vec{E}## is radially out.

PS: Questions like this should be posted in the homework forum!
 

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