Nonuniform Spherical Charge Density
Q_enclosed(r) = integral_0^r rho(r')4 pi r'^2 dr', and E(r) = Q_enclosed(r)/(4 pi epsilon_0 r^2). Normalize rho using the stated total charge when needed.
Why this shows up in the exam
Power-law density profiles · Piecewise nuclear charge density · Normalizing an unknown density constant
Learn the idea
For spherical rho(r), integrate enclosed charge first and then apply Gauss's law. Shells at different radii can carry different density. Spherical symmetry still makes E radial, but the enclosed charge is now an integral rather than a simple volume fraction.
🧠 Memory hook: Density integrates to charge; charge divided by r squared gives field.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- Q_enclosed(r) = integral₀^r 4 pi r'² rho(r') dr' — enclosed charge
- E(r) = Q_enclosed(r)/(4 pi epsilon₀ r²) — radial field from Gauss's law
How to approach it
- 1Normalize the density if required
- 2Integrate only to the observation radius
- 3Apply Gauss and test centre/surface limits
Common slip-ups that cost marks
- •Using rho(r) times the whole volume
- •Integrating to R for an interior field point
- •Forgetting piecewise limits
🌟 That's the whole idea — you've got this. Try the practice set below; every question you attempt makes it stick a little harder.
Original chapter practice
Original questions for this chapter, not past-paper questions or an exact mapping to this individual concept.
Two point charges 1 microC and 2 microC are 1 m apart in vacuum. Take k = 9 x 10^9 SI. Find the force magnitude.
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