Spatially Varying Dielectric
For planar plates with epsilon(x) varying only along the field and negligible fringing, electric displacement is uniform and the total voltage is the integral of D/epsilon(x).
Why this shows up in the exam
graded dielectrics · K(x)=K_0+lambda x · continuous layered media
Learn the idea
A dielectric varying along the field must be integrated as differential series layers. When permittivity changes continuously from one plate to the other, tiny slices each take part of the voltage drop; their reciprocal capacitances add.
🧠 Memory hook: Variation along field: integrate 1/K.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- 1/C = (1/A) integral₀^d dx/epsilon(x) — variation along field
- C/C₀ = d/[integral₀^d dx/K(x)] — C₀=epsilon₀ A/d
How to approach it
- 1Identify the variation direction
- 2Write differential reciprocal capacitance
- 3Integrate and compare with C₀
Common slip-ups that cost marks
- •Averaging K arithmetically along thickness
- •Treating every slice as parallel
- •Forgetting integration limits and dimensions
🌟 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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