Explanation Given data: The volume charge density is ρ v = 50 r e − r C / m 3 . Calculation: The required diagram is drawn as shown in Figure 1. Write the standard expression of differential form of Gauss’s law. ∇ ⋅ D = ρ v (1) Here, D is the electric flux. ∇ is the gradient operator. ρ v is the volume charge density. Write the divergence of electric flux in cylindrical coordinate system. ∇ ⋅ D = 1 r ∂ ( r D r ) ∂ r + 1 r ∂ ( D θ ) ∂ θ + ∂ ( D z ) ∂ z Here, r is the radius of the cylinder. The volume charge density ρ v is only function of r . Thus, electric flux is also the function of r only that is, D = r ^ D r (2) Here, D r is the radial component. The divergence of electric flux is rewritten as, ∇ ⋅ D = 1 r ∂ ( r D r ) ∂ r Substitute 1 r ∂ ( r D r ) ∂ r for ∇ ⋅ D and 50 r e − r for ρ v in the equation (1). 1 r ∂ ( r D r ) ∂ r = 50 r e − r ∂ ( r D r ) ∂ r = 50 r 2 e − r Integrate the above equation on both side from r = 0 to r = r

7th Edition

ISBN: 8220100663659

Author: ULABY

Publisher: PEARSON

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Chapter 4, Problem 26P

To determine

The electric flux D for the given condition.

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Chapter 4, Problem 25P

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Chapter 4 Solutions

EBK FUNDAMENTALS OF APPLIED ELECTROMAGN

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