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Ch 24: Gauss' Law
Knight Calc - Physics for Scientists and Engineers 5th Edition
Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Non è quello che usi tu?Cambia libro di testo
Tutti i libri di testoKnight Calc 5th EditionCh 24: Gauss' LawProblema 61b
Capitolo 24, Problema 61b

A spherical ball of charge has radius R and total charge Q. The electric field strength inside the ball (r ≤ R ) is E(r)=r4Emax/R4E(r)=r^4E_{max}/R^4. Find an expression for the volume charge density ρ(r) inside the ball as a function of r.

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Step 1: Recall the relationship between the electric field and the charge density. Gauss's law in differential form states that the divergence of the electric field is proportional to the charge density: ∇·E = ρ/ε₀, where ρ is the volume charge density and ε₀ is the permittivity of free space.
Step 2: Write the expression for the electric field inside the ball, which is given as E(r) = (r⁴ Eₘₐₓ) / R⁴. Since the electric field is radially symmetric, we can use the spherical form of Gauss's law: (1/r²) ∂(r²E(r))/∂r = ρ(r)/ε₀.
Step 3: Compute the derivative of r²E(r) with respect to r. Start by multiplying the electric field E(r) by r²: r²E(r) = r² * (r⁴ Eₘₐₓ / R⁴) = r⁶ Eₘₐₓ / R⁴. Now, differentiate this expression with respect to r: ∂(r²E(r))/∂r = ∂(r⁶ Eₘₐₓ / R⁴)/∂r = 6r⁵ Eₘₐₓ / R⁴.
Step 4: Substitute the derivative into Gauss's law. Using (1/r²) ∂(r²E(r))/∂r = ρ(r)/ε₀, replace ∂(r²E(r))/∂r with 6r⁵ Eₘₐₓ / R⁴: (1/r²) * (6r⁵ Eₘₐₓ / R⁴) = ρ(r)/ε₀.
Step 5: Simplify the expression to find ρ(r). Combine terms: ρ(r) = ε₀ * (6r³ Eₘₐₓ / R⁴). This is the final expression for the volume charge density inside the ball as a function of r.

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Electric Field

The electric field is a vector field that represents the force exerted by an electric charge on other charges in its vicinity. Inside a charged spherical object, the electric field varies with distance from the center, and in this case, it is given by E(r) = r⁴ Eₘₐₓ / R⁴, indicating that the field strength increases with the fourth power of the radius.
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Intro to Electric Fields

Volume Charge Density

Volume charge density (ρ) is defined as the amount of electric charge per unit volume within a specified region. It is crucial for understanding how charge is distributed within a volume, and can be derived from the electric field using Gauss's law, which relates the electric field to the charge enclosed within a Gaussian surface.
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Gauss's Law

Gauss's Law states that the total electric flux through a closed surface is proportional to the enclosed electric charge. This principle is fundamental in electrostatics, allowing us to relate the electric field to charge distributions, particularly in symmetric situations like spherical charge distributions, which simplifies the calculation of electric fields and charge densities.
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Gauss' Law
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