Effective Resistivity of Composite Conductors
For equal-length parallel wires, conductivities are area-weighted. For equal-area series layers, resistivities are length-weighted. Other geometries must be reduced from R = rho L/A.
Why this shows up in the exam
Composite cables · Effective conductivity questions · Mixed-material rods
Learn the idea
An effective material property depends on whether different materials carry current in series or parallel. Layered conductors behave like resistor combinations. Along the length, voltage drops add; side-by-side, currents add. Only after finding total resistance should an effective resistivity be defined for the chosen overall geometry.
🧠 Memory hook: Combine actual resistances first; name rho-effective last.
Get this one clearly and it pays off every single time it shows up in the paper. 🎯
Formulas & facts to keep ready
- sigma_eff = sum(sigma_i A_i)/sum A_i — parallel equal-length conductors
- rho_eff = sum(rho_i L_i)/sum L_i — series equal-area layers
- rho_eff = R_eq A_total/L_total — definition for the specified equivalent sample
How to approach it
- 1Draw current direction through each material
- 2Compute or combine conductances/resistances
- 3Convert back to the requested effective property
Common slip-ups that cost marks
- •Averaging resistivities without geometric weights
- •Using the series rule for parallel wires
- •Confusing effective resistance with effective resistivity
🌟 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.
A cell of emf 6 V and internal resistance 1 ohm is connected to a 2 ohm resistor. Find the circuit current.
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