Mathematical routines (MR)
Capacitance of a coaxial cable
- Units 8, 9 and 10
- 10 points
- About 22 minutes
You can use a calculator on this question, just like on exam day.
A multipart problem where you use math, including calculus, to analyze a physical situation. You draw or sketch a representation such as a diagram or a graph, derive an equation using letters, calculate a numerical answer with units, and make a claim or prediction and justify it with physics reasoning. On the exam: Question 1 of 4. Section II has 4 questions in 95 minutes (50% of the exam score), one of each type in this order; calculator and equation sheet allowed. The CED suggests 20–25 minutes.
The question
A long coaxial cable is made of a solid inner conducting wire of radius a, surrounded by a thin, coaxial cylindrical conducting shell of radius b. The space between them is completely filled with an insulating material of dielectric constant κ. The cable has length ℓ, which is much greater than b, so ignore edge effects. The inner wire has charge +Q, and the outer shell has charge −Q.
Suggested time: 22 minutes
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Part (a)
2 pointsUse Gauss’s law to derive an expression for the magnitude of the electric field in the insulating material at a distance r from the axis, where a < r < b.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
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Part (b)
2 pointsDerive an expression for the magnitude of the potential difference between the inner wire and the outer shell.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (c)
1 pointDerive an expression for the capacitance of the cable.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (d)
2 pointsThe cable has a = 0.50 mm, b = 2.0 mm, ℓ = 10 m, and κ = 2.3. It is connected to a 12 V battery. Calculate (i) the capacitance of the cable and (ii) the magnitude of the charge on the inner wire.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
Part (e)
1 pointCalculate the maximum magnitude of the electric field in the insulating material, and state where it occurs.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
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Part (f)
2 pointsWith the battery still connected, the insulating material is slid out of the cable so that the gap is filled with air (κ ≈ 1). A student claims that the charge on the inner wire will stay the same because charge is conserved. Do you agree or disagree? Justify your answer.
Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).
0 / 2,500 characters
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