Potential Graphs, Poisson Equation, and Charge Density
In a homogeneous linear medium, electrostatic potential satisfies div(grad V)=-rho/epsilon; charge-free regions satisfy Laplace equation.
Why this shows up in the exam
reading V-r graphs · finding volume charge density · checking allowed charge-free potentials
Learn the idea
Slopes give field and curvature gives charge density. A potential graph stores more than voltage: its first spatial derivative gives field, and its curvature reveals where charge is present.
🧠 Memory hook: Slope maps field; curvature maps charge.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- nabla² V = -rho/epsilon — homogeneous linear medium
- nabla² V = 0 — charge-free region
- E_x = -dV/dx — one-dimensional graph or function
How to approach it
- 1Identify symmetry and coordinates
- 2Differentiate with the correct Laplacian
- 3Check units and region boundaries
Common slip-ups that cost marks
- •Using first derivative for charge density
- •Applying Cartesian second derivative to a radial function
- •Ignoring discontinuities that represent surface charge
🌟 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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