Passive Sign Convention and Power: Practice Problems with Answers

S1Prefixes

A sensor’s output current is 0.000047 A. Which is the way engineers write it, with the prefix chosen so the number lies between 1 and 1000?

Answer

a)0.047 mAb)47 µAc)47 nAd)$4.7 \times 10^{-5}$ mA

S2Your generator subscriptionTwo-tier

Your home has a 5 A generator subscription at 220 V. Treat each rating as the power the appliance draws: kettle 2000 W, washing machine 500 W, heater fan 300 W, fridge 150 W, TV 100 W, laptop 65 W, all LED lights together 60 W. Which set can run at the same time without tripping the breaker?

Answer

a)Kettle and LED lightsb)Washing machine, heater fan, fridge, TV, laptop and LED lightsc)Washing machine, fridge, TV, laptop and LED lightsd)Kettle alone

Reason

i)The subscription limits energy per month, so any set works if used briefly.ii)The subscription limits power, $P = VI = 1100$ W at any moment.iii)The subscription limits current, so the voltage does not matter.iv)Only the largest appliance counts.

S3Charge from a current graphPredict first

The graph shows the current entering an element (zero outside the times shown).

Predict first: is the total charge more or less than 4 mA × 6 s = 24 mC?

a)More than 24 mCb)Less than 24 mCc)Exactly 24 mC

How much charge enters the element in total?

a)24 mCb)16 mCc)22 mCd)18 mC

S4Current from chargeTwo-tier

The charge entering an element is $q(t) = 4(1 - e^{-250t})$ mC for $t \ge 0$, and zero before. What is the current at $t = 0^+$?

Answer

a)0b)4 mAc)1 Ad)250 mA

Reason

i)$q(0) = 0$, so $i(0) = 0$.ii)Current is the charge divided by the time.iii)Current is the slope $dq/dt$, which is largest at $t = 0$.iv)Current is the final charge, 4 mC, per second.

S5What a negative current meansTwo-tier

Through the element, positive charge actually flows:

Answer

a)from a to bb)from b to ac)nowhere, because a negative current means no flowd)cannot tell without the voltage

Reason

i)The arrow is only a reference; a negative value means the flow is opposite to the arrow.ii)A negative current means electrons flow instead of charge.iii)Current always flows from + to −.iv)The arrow shows the true direction, whatever the sign.

Follow-up: in which direction do the electrons move?

a)from a to bb)from b to a

S6What a negative voltage means

Which statement is true?

Answer

a)Terminal b is 12 V higher than terminal a.b)Terminal a is 12 V higher than terminal b.c)Terminals a and b are at the same potential.d)The element must be delivering power.

S7Predict, then computePredict first

Four elements are connected between the same two wires. Their reference values are:

Element	A	B	C	D
$v$ (V)	12	−12	−12	12
$i$ (A)	3	−5	−4	2

Predict first, with no numbers: for which elements does the reference arrow enter the + terminal?

a)A and Bb)B and Dc)C and Dd)A and C

Which elements absorb power?

a)A and Bb)B and Dc)C and Dd)A and C

Total power delivered

W

S8One elementTwo-tier

The element:

Answer

a)absorbs 20 Wb)delivers 20 Wc)absorbs 10 Wd)delivers 40 W

Reason

i)The current arrow enters the − terminal, so $p = -vi$.ii)$v$ and $i$ have opposite signs, so $p$ is negative.iii)Current flows into the element, so it absorbs.iv)$v$ is positive, so it absorbs.

S9Same element, different arrowsTwo-tier

Student X draws the current arrow into the + terminal of an element and finds $v = 5$ V, $i = 2$ A. Student Y analyses the same element but draws the current arrow the other way, keeping the same voltage polarity. What does Y get?

Answer

a)$i = 2$ A, $p = -10$ W (delivered)b)$i = -2$ A, $p = -10$ W (delivered)c)$i = -2$ A, $p = 10$ W (absorbed)d)Y cannot get a valid answer with that arrow.

Reason

i)Reversing an arrow changes the sign of the number, not what the element physically does.ii)Reversing the arrow reverses the physical current.iii)Only arrows that enter the + terminal are allowed.iv)Power has no sign, so both get 10 W.

S10Charging a battery

A 12 V car battery is charged at a constant 5 A for 2 h. How much energy is delivered to the battery?

Answer

a)120 Jb)7.2 kJc)432 kJd)432 J

Follow-up: the same energy in kWh

kWh

Follow-up: while charging, the charger current enters the battery’s…

a)+ terminalb)− terminal

S11Your power bankTwo-tier

A power bank is rated 10,000 mAh at 3.7 V. It charges a phone through a 5 V, 3 A output. Ignoring losses, how long can it supply that output?

Answer

a)3.3 hb)2.5 hc)0.8 hd)12.3 h

Reason

i)mAh measures charge; when the voltage changes, energy (Wh) is what carries over.ii)mAh measures energy, so the time is 10 Ah ÷ 3 A.iii)The phone’s voltage does not matter.iv)The time is the energy in Wh divided by the current in A.

S12Energy from time-varying power

For $t \ge 0$, an element has $v = 50e^{-200t}$ V and $i = 4e^{-200t}$ A, with the current entering the + terminal. What total energy is delivered to the element from $t = 0$ to $\infty$?

Hint: First write $p(t)$. How fast does $p$ decay compared with $v$ and $i$?

Answer

a)0.5 Jb)1 Jc)200 Jd)0.25 J

S13Power balanceTwo-tier

A circuit has five elements. Using the passive sign convention, four of the powers are $p_1 = -60$ W, $p_2 = +25$ W, $p_3 = +18$ W and $p_4 = -8$ W. What is $p_5$?

Answer

a)9 W absorbedb)25 W deliveredc)111 W absorbedd)25 W absorbed

Reason

i)The powers in a circuit add up to zero.ii)Only the delivered powers need to balance.iii)Absorbed power is always larger than delivered power.iv)$p_5$ is the sum of the other four.

S14Spot the errorSpot the error

The worked solution below contains the kind of mistake AI chat tools most often make on circuit diagrams.

The 3 A source drives a clockwise current of 3 A through elements 1 and 2.
Element 1: $p_1 = (9)(3) = 27$ W, absorbed.
Element 2: the current enters the + terminal, so $p_2 = (15)(3) = 45$ W, absorbed.
KVL clockwise, drops positive: $-V - 9 + 15 = 0$, so $V = 6$ V.
Source: the current leaves its + terminal, so $p_s = -(6)(3) = -18$ W, delivered.
All powers have been found, so the solution is complete.

b) The corrected value of $p_1$

W

c) With the corrected values, the total power delivered

W

S15Work backwards

The current arrow of an element enters its + terminal and $i = 2$ A. What must $v$ be for the element to deliver 30 W?

Answer

a)15 Vb)−15 Vc)60 Vd)−60 V

S16When does circuit theory apply?

Use $\lambda = c/f$ with $c = 3 \times 10^8$ m/s and the one-tenth rule. Which systems can be analysed as lumped circuits?
(i) a 30 km distribution feeder at 50 Hz (ii) a 10 cm Wi-Fi board at 2.4 GHz (iii) a 20 cm audio amplifier at 20 kHz

Answer

a)(i) onlyb)(i) and (iii)c)all threed)(ii) only