Aim for FivePhysics C: E&M

Mathematical routines (MR)

Charging a capacitor through a resistor

  • Unit 11
  • 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

An uncharged capacitor of capacitance C is connected in series with a resistor of resistance R, an open switch, and an ideal battery of emf ε. The switch is closed at time t = 0.

Suggested time: 22 minutes

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Part (a)

2 points

Apply Kirchhoff’s loop rule to write a differential equation for the charge q on the capacitor as a function of time t.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (b)

2 points

Solve your differential equation to derive an expression for q(t) in terms of ε, R, C, and t.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (c)

2 points

The resistance is 50 kΩ, the capacitance is 200 μF, and the emf is 12 V. Calculate the time at which the potential difference across the capacitor is 9.0 V.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (d)

2 points

After a long time, the capacitor is fully charged. Calculate (i) the total energy supplied by the battery during the charging and (ii) the energy stored in the capacitor.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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Part (e)

2 points

The resistor is replaced with one of resistance 2R, the capacitor is fully discharged, and the switch is closed again. A student claims that, because the resistance is larger, more energy will be converted to thermal energy in the resistor during the full charging process. Do you agree or disagree? Justify your answer.

Type math plainly, like x^2, sqrt(x) or (x+1)/(x−1).

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