Open practice · AI allowed

Practice sheet: Chapter 6, Inductors and Capacitors

This sheet is not marked for correctness, and AI tools are allowed. The device-free quiz at the start of week 12, session 1 (Tue 5 Jan), is built from twins of these questions, so make sure you can do each one on your own.

Two-tier items ask for an answer and a reason. Get both right; on the quiz each earns half the marks.
Predict first: where asked, make your prediction before you calculate.
Confidence: mark each question Sure or Unsure. It is not marked; it shows you what to revise.
Conventions: unless a figure shows otherwise, each element’s current is referenced in the direction of its voltage drop (passive sign convention), so $v = L\,\tfrac{di}{dt}$ and $i = C\,\tfrac{dv}{dt}$. An initial current or voltage that points against the reference is negative. Check every answer: put $t = 0$ into your result, and differentiate it back.

Prefer paper? Open the printable PDF.

Which questions practise which skill

1Find an inductor’s voltage from its current, and its current from its voltage and initial current S2 S3 S10 S12 S16
2Do the same for a capacitor’s current and voltage S5 S6 S9 S11 S15
3Find the power and stored energy, and say when the element stores or returns energy S1 S2 S3 S4 S5 S6 S7 S8 S9 S14
4State the dc behaviour of each element and which quantity cannot change instantly S1 S2 S4 S5
5Combine inductors or capacitors in series and parallel, with their initial currents or voltages S7 S8 S9 S10 S11 S13 S14 S15
6Find the energy delivered to a black box and the energy trapped in the elements S10 S11 S13

S1An inductor in a dc circuitTwo-tier

A 2 H inductor in a dc circuit that has been on for a long time carries a constant 3 A. What is the voltage across it?

Answer

a)6 Vb)0 Vc)1.5 Vd)3 V

Reason

i)The voltage is $v = Li$.ii)The voltage is $v = L\,di/dt$, and the current is not changing.iii)The voltage is $v = i/L$.iv)The voltage is the stored energy divided by the current, $w/i = Li/2$.

Follow-up: how much energy does the inductor store?

S2Voltage from current

As in Nilsson Example 6.1. The current in a 0.5 H inductor is 0 for $t < 0$ and $i = 8te^{-10t}$ A for $t \ge 0$.

a) When is the current largest, and what is it? The time

a) The largest current

b) Find $v(t)$ for $t > 0$, in the form $v = K_1 e^{-10t}(1 - K_2 t)$ V: $K_1$

b) $K_2$

1/s

c) When does the voltage change polarity?

d) Does the voltage change instantly at $t = 0$?

a)Yesb)No

d) Does the current?

a)Yesb)No

e) What is the largest energy the inductor stores?

S3Current from voltage

As in Nilsson Example 6.2. A 2 H inductor carries $i(0) = -1$ A. For $t > 0$ the voltage across it is $v = 12e^{-3t}$ V.

a) Find $i(t)$ for $t \ge 0$, in the form $i = K_1 + K_2 e^{-3t}$ A: $K_1$

a) $K_2$

b) When is the current zero?

c) In what interval does the inductor deliver energy? Use the sign of $p = vi$.

a)$0 < t < 0.231$ sb)$t > 0.231$ sc)All $t > 0$d)Never

d) Find the stored energy at $t = 0$

d) … and as $t \to \infty$

d) Equal values: is that a contradiction?

a)Yesb)No

S4A capacitor in a dc circuitTwo-tier

A 5 µF capacitor in a dc circuit that has been on for a long time has a constant 12 V across it. What current flows in it?

Answer

a)60 µAb)30 µAc)0 Ad)$2.4 \times 10^6$ A

Reason

i)The current is the stored energy divided by the voltage, $w/v = Cv/2$.ii)The current is $i = Cv$.iii)The current is $i = v/C$.iv)The current is $i = C\,dv/dt$, and the voltage is not changing.

Follow-up: how much energy does the capacitor store?

µJ

S5Current from voltage

As in Nilsson Example 6.4. The voltage across a 4 µF capacitor is 0 for $t < 0$ and $v = 20(1 - e^{-500t})$ V for $t \ge 0$.

a) Find $i(t)$ for $t > 0$, in the form $i = K e^{-500t}$ mA: $K$

b) Does the current change instantly at $t = 0$?

a)Yesb)No

b) Does the voltage?

a)Yesb)No

c) Find the energy stored as $t \to \infty$.

d) When is the power into the capacitor largest, and what is it? The time

d) The largest power

S6A current pulsePredict first

As in Nilsson Example 6.5. A 0.5 µF capacitor has $v(0) = 5$ V. A current of 2 mA flows into its + terminal from $t = 0$ to $t = 5$ ms, and zero afterwards.

a) Find $v(t)$ for $0 \le t \le 5$ ms, in the form $v = K_1 + K_2 t$ V: $K_1$

a) $K_2$

V/s

b) What is $v$ for $t > 5$ ms?

b) Why does it not return to 5 V?

Work this out on paper, then compare with the explanation.

c) How much energy does the pulse deliver to the capacitor?

µJ

d) Predict first: repeat (a)–(c) with a −2 mA pulse. $v$ for $t > 5$ ms

d) The energy the −2 mA pulse delivers to the capacitor

µJ

d) When does $v$ pass through zero?

S7Equivalent inductance with initial currents

As in Nilsson Example 6.6.

a) Find $L_{eq}$ between a and b.

b) Find the initial current in $L_{eq}$, entering at a.

c) The 15 mH and 5 mH carry the current found in (b). Find the energy stored in the four inductors

c) … and in $L_{eq}$

c) Where is the difference?

a)In the current that circulates around the 30 mH and 60 mH pairb)In the 15 mH and 5 mH, which carry the full 3 Ac)Nowhere: the two totals must be equal, so one of them is wrong

S8Equivalent capacitance with initial voltages

As in Nilsson Example 6.7.

a) Find $C_{eq}$ between a and b.

µF

b) Find the initial voltage $v_{ab}$ across $C_{eq}$.

c) What is the initial voltage across the 8 µF, + at the top?

c) Find the energy stored in the four capacitors

µJ

c) … and in $C_{eq}$

µJ

S9Parallel capacitors

As in booklet Problem 6.9. A 0.4 µF and a 1.6 µF capacitor are in parallel. For $t \ge 0$ the voltage across them is $v = 50e^{-200t} + 30$ V. $i_1$ is the current in the 0.4 µF and $i_2$ the current in the 1.6 µF, each referenced into the + terminal.

a) Find $C_{eq}$.

µF

b) Find $i_1(t)$ and $i_2(t)$, in the form $i_1 = K_1 e^{-200t}$ and $i_2 = K_2 e^{-200t}$: $K_1$

b) $K_2$

c) Check that $i_1 + i_2 = C_{eq}\,dv/dt$. How do the two currents compare with the two capacitances?

a)They divide in the ratio of the capacitances: $i_1 : i_2 = 1 : 4$.b)They divide in the inverse ratio of the capacitances: $i_1 : i_2 = 4 : 1$.c)They are equal, because both capacitors have the same voltage.

d) How much energy do the capacitors release between $t = 0$ and $t \to \infty$?

S10Inductors and a black box

As in booklet Problem 6.8. Just before the switch opens, $i_1(0) = -9$ A and $i_2(0) = 3$ A. For $t > 0$, $v = 36e^{-3t}$ V.

a) Find $L_{eq}$

a) … and $i(0^+)$, the box current just after the switch opens

b) Find $i(t)$, $i_1(t)$ and $i_2(t)$ for $t \ge 0$, each in the form $K_1 + K_2 e^{-3t}$ A. For $i(t)$: $K_1$

b) For $i(t)$: $K_2$

b) For $i_1(t)$: $K_1$

b) For $i_1(t)$: $K_2$

b) For $i_2(t)$: $K_1$

b) For $i_2(t)$: $K_2$

c) Find the initial energy stored in the inductors.

d) Find the energy delivered to the black box

d) … and the energy trapped in the inductors

S11Capacitors and a black box

As in booklet Problem 6.10. At $t = 0$, $v_1(0) = 20$ V and $v_2(0) = -8$ V. For $t > 0$, $i = 1.2e^{-50t}$ mA.

a) Find $C_{eq}$

µF

a) … and $v_o(0)$

b) Find $v_1(t)$, $v_2(t)$ and $v_o(t)$ for $t \ge 0$, each in the form $K_1 + K_2 e^{-50t}$ V. For $v_1(t)$: $K_1$

b) For $v_1(t)$: $K_2$

b) For $v_2(t)$: $K_1$

b) For $v_2(t)$: $K_2$

b) For $v_o(t)$: $K_1$

b) For $v_o(t)$: $K_2$

c) Find the initial energy stored in the capacitors.

µJ

d) Find the energy delivered to the black box

µJ

d) … and the energy trapped in the capacitors

µJ

S12Spot the errorSpot the error

The worked solution below contains the kind of direction slip AI chat tools often make. A 4 H inductor has its current $i$ referenced in the direction of its voltage drop $v$. At $t = 0$ the current is 2 A, flowing against the reference arrow. For $t > 0$, $v = 24e^{-2t}$ V.

$i(t) = (1/L)\int v\,dt + i(0)$, integrating from 0 to $t$.
$i(0) = 2$ A.
The integral of $24e^{-2t}$ from 0 to $t$ is $12(1 - e^{-2t})$.
$i(t) = (12/4)(1 - e^{-2t}) + 2 = 5 - 3e^{-2t}$ A.
Check: $i(0) = 2$ A and $4\,di/dt = 24e^{-2t}$ V, so the answer is right.

b) Correct it and find $i(t)$. The corrected $i(0)$

b) $i(t)$ in the form $K_1 + K_2 e^{-2t}$ A: $K_1$

b) $K_2$

c) Line 5 calls itself a check. Why did it not catch the error?

Work this out on paper, then compare with the explanation.

d) When does the current reverse direction?

S13Spot the error in an energy balanceSpot the error

A 2 H and an 8 H inductor are in parallel, with $i_1(0) = 6$ A and $i_2(0) = -1$ A, both referenced downward; until $t = 0$ a switch across their terminals carries the total current. At $t = 0$ the switch opens and the current flows into a black box, which draws current until the total current is zero.

$L_{eq} = (2)(8)/(2 + 8) = 1.6$ H.
The initial stored energy is $(1/2)(2)(6)^2 + (1/2)(8)(1)^2 = 36 + 4 = 40$ J.
The box draws current until the inductors are empty, so it receives all the stored energy.
The energy delivered to the box is 40 J.

a) Which is the first wrong line?

a)Line 1b)Line 2c)Line 3d)Line 4

b) Find the energy delivered to the box. (Use $L_{eq}$ and the total initial current.)

c) Find $i_1$ after a long time

c) … $i_2$ after a long time

c) … and the energy trapped

S14A supercapacitor bankPredict first

A supercapacitor cell is rated 3000 F at 2.7 V.

a) How much energy does one cell store at its rated voltage?

b) Six cells are connected in series for a 16.2 V bank. Find $C_{eq}$

b) … and the energy the bank stores at 16.2 V

b) Compare with six times the energy of one cell.

a)The bank stores exactly six times the energy of one cell.b)The bank stores less than six times the energy of one cell.c)The bank stores more than six times the energy of one cell.

c) For how long could the bank supply 200 W if all of that energy could be used?

d) Predict first: two such banks are put in parallel. What happens to $C_{eq}$ and to the stored energy?

a)$C_{eq}$ doubles and the stored energy doubles.b)$C_{eq}$ doubles and the stored energy stays the same.c)$C_{eq}$ halves and the stored energy halves.d)$C_{eq}$ doubles and the stored energy quadruples.

d) The energy the two banks store at 16.2 V

S15Capacitive touch screens

As in Nilsson’s Practical Perspective for this chapter. Each electrode of a touch screen has a parasitic capacitance $C_p = 20$ pF to ground. A finger adds $C_t = 5$ pF at the point of touch.

a) What capacitance does the controller see at the touched electrode?

a) By what percentage did it change?

b) A self-capacitance screen is touched at (X1, Y2) and (X3, Y0) at the same time. Which four grid points look touched?

a)(X1, Y2), (X3, Y0), (X1, Y0) and (X3, Y2)b)(X1, Y2), (X3, Y0), (X2, Y1) and (X0, Y3)

b) Which two are ghosts?

a)(X1, Y0) and (X3, Y2)b)(X1, Y2) and (X3, Y0)c)(X2, Y1) and (X0, Y3)

c) Why does a mutual-capacitance screen not produce ghost points?

Work this out on paper, then compare with the explanation.

d) The controller charges an electrode with a constant 1 µA. How long does the voltage take to reach 1 V when untouched

µs

d) … and when touched?

µs

S16Check an answer without solving again

A 0.25 H inductor carries $i(0) = 3$ A. For $t > 0$, $v = 10e^{-20t}$ V. Two classmates report $i(t)$ for $t \ge 0$. Student 1: $i = 5 - 2e^{-20t}$ A. Student 2: $i = 1 + 2e^{-20t}$ A. Use $i(0)$ and $v = L\,di/dt$ to decide which student is right.

Answer

a)Student 1b)Student 2c)Bothd)Neither