recall and use Q = It

Cambridge International AS & A Level Physics 9702 – Electric Current ( $Q = I t$ )

1. Syllabus Reference

Topic 9.1 – Electric current. Learning objective: Recall and use the relationship $Q = I t$ to calculate charge, current or time in a variety of contexts.

2. Learning Outcomes

• Identify the three fundamental electrical quantities (charge $Q$, current $I$, time $t$) and their SI units.
• State, rearrange and apply the three forms of the equation $Q = I t$.
• Convert between common prefixes (µ, m, k) and between time units (s, min, h).
• Explain the microscopic origin of current using $I = n A q v_{d}$ and give typical values for metals and electrolytes.
• Recognise the quantisation of charge and calculate the number of charge carriers involved in a given charge.
• Interpret simple circuit diagrams and recognise the standard symbols for current sources, batteries, ammeters, voltmeters, switches, resistors and lamps.
• Identify typical sources of error in current measurements and state the key safety precautions when working with electric circuits.

3. Quick‑Reference Box

Form	Expression	When to use
$Q$	$Q = I\,t$	Find charge when current and time are known.
$I$	$I = \dfrac{Q}{t}$	Find current when charge transferred and time are given.
$t$	$t = \dfrac{Q}{I}$	Find the duration of a current flow.

Always keep units consistent (A, C, s). Convert minutes or hours to seconds before substituting.

4. Physical Quantities & Units (Course‑wide reminder)

Quantity	Symbol	SI Unit	Common prefixes
Mass	m	kilogram (kg)	g = 10⁻³ kg, mg = 10⁻⁶ kg
Length	l	metre (m)	cm = 10⁻² m, mm = 10⁻³ m
Time	t	second (s)	min = 60 s, h = 3600 s
Electric current	I	ampere (A)	mA = 10⁻³ A, µA = 10⁻⁶ A
Electric charge	Q	coulomb (C)	mC = 10⁻³ C, µC = 10⁻⁶ C

5. Key Concepts

• Electric charge ($Q$) – amount of electricity. $1\;\text{C}=6.242\times10^{18}$ elementary charges ($e$).
• Elementary charge ($e$) – magnitude of charge on a single electron or proton, $e = 1.602\times10^{-19}\;\text{C}$.
• Electric current ($I$) – rate of charge flow past a point: $I = \dfrac{Q}{t}$ (unit A = C s⁻¹).
• Time ($t$) – duration for which the current flows.
• Microscopic view: $$I = n A q v_{d}$$ where
  • $n$ – number density of charge carriers (m⁻³). Typical values: $n\approx8.5\times10^{28}\;\text{m}^{-3}$ for copper, $n\approx10^{20}\;\text{m}^{-3}$ for a 0.1 M NaCl solution.
  • $A$ – cross‑sectional area of the conductor (m²).
  • $q$ – charge on each carrier (for electrons $q = e$).
  • $v_{d}$ – drift velocity (m s⁻¹), usually $10^{-4}$–$10^{-3}$ m s⁻¹ in metals.
• Quantisation of charge: charge is transferred in integer multiples of $e$. The number of carriers transferred is $N = Q/e$.

6. Derivation & Rearrangement

Starting from the definition of current:

$$I = \frac{Q}{t}$$

Multiplying both sides by $t$ gives the most frequently used form:

$$Q = I\,t$$

If the current varies with time, the total charge is the time‑integral of the current:

$$Q = \int I(t)\,dt$$

7. Units, Prefixes & Conversions

Quantity	Symbol	SI unit	Useful conversions
Charge	Q	C	1 C = 10⁶ µC = 10³ mC = 6.242×10¹⁸ e
Current	I	A	1 A = 10³ mA = 10⁶ µA
Time	t	s	1 min = 60 s, 1 h = 3600 s

8. Standard Circuit Symbols (Topic 9.1)

Component	Symbol	Function
Current source (arrow)		Provides a steady current.
Battery (cell)		Source of emf; polarity indicated by longer line.
Ammeter		Measures current (connected in series).
Voltmeter		Measures potential difference (connected in parallel).
Switch		Opens or closes a circuit.
Resistor (or lamp)		Imposes resistance; the zig‑zag symbol is also used for a lamp.

9. Safety & Practical Tips

• Always wear appropriate personal protective equipment (PPE): insulated gloves, safety glasses, and closed shoes.
• Never work on a live circuit; disconnect the power supply and isolate the part being measured.
• Maximum safe current through dry human skin is roughly 0.01 A. Treat any higher current as hazardous.
• Use fuses or circuit breakers that are rated for the expected current; they protect by melting after a specified charge (e.g., 10 C).
• When converting units, write the conversion factor explicitly to avoid a factor‑of‑1000 error.
• If the current is not constant, record a current‑time graph and use the area under the curve (integration) to obtain total charge.

10. Sources of Error (Experimental Skills – AO3)

• Instrumental errors: internal resistance of an ammeter, limited resolution of the display.
• Connection errors: loose contacts, reversed polarity, or unintentional short circuits.
• Reading errors: parallax when reading analog meters.
• Timing errors: human reaction time when using a stopwatch; mitigate by using digital timers.
• Environmental factors: temperature changes affecting resistance and thus current.

11. Derivation of the Microscopic Formula (Optional)

In a conductor of cross‑section $A$, $n$ carriers per cubic metre each carry charge $q$. In time $t$, a carrier travels a distance $v_{d}t$, sweeping out a volume $A\,v_{d}t$. The number of carriers that pass a given cross‑section is $nA v_{d}t$, so the total charge is

$$Q = (nA v_{d}t)q \quad\Longrightarrow\quad I = \frac{Q}{t}=nA q v_{d}.$$

12. Worked Examples

Example 1 – Direct use of $Q = I t$

Problem: A 2.0 A lamp is switched on for 3.5 minutes. Find the total charge that passes through the lamp.

1. Convert time: $t = 3.5\;\text{min}\times60\;\frac{\text{s}}{\text{min}} = 210\;\text{s}$.
2. Apply $Q = I t$: $Q = (2.0\;\text{A})(210\;\text{s}) = 420\;\text{C}$.

Result: $420\;\text{C}$ of charge flow.

Example 2 – Microscopic view (drift velocity)

Problem: A copper wire of cross‑section $A = 1.0\;\text{mm}^2$ carries $5.0\;\text{A}$. Copper has $n = 8.5\times10^{28}\;\text{m}^{-3}$. Find $v_{d}$.

1. Convert area: $A = 1.0\;\text{mm}^2 = 1.0\times10^{-6}\;\text{m}^2$.
2. Use $v_{d}=I/(nA e)$: $$v_{d}= \frac{5.0}{(8.5\times10^{28})(1.0\times10^{-6})(1.602\times10^{-19})}\approx3.7\times10^{-4}\;\text{m s}^{-1}.$$

The drift speed is only a few tenths of a millimetre per second.

Example 3 – Quantisation of charge

Problem: How many electrons pass a point when $5.0\;\text{C}$ of charge flows?

Number of electrons $N = Q/e = \dfrac{5.0}{1.602\times10^{-19}} \approx 3.1\times10^{19}$ electrons.

Example 4 – Variable current (integration)

Problem: A current varies with time as $I(t)=2.0t$ A, where $t$ is in seconds, from $t=0$ to $t=3\;$s. Find the total charge transferred.

$$Q=\int_{0}^{3}2.0t\,dt = 2.0\left[\frac{t^{2}}{2}\right]_{0}^{3}=2.0\left(\frac{9}{2}\right)=9\;\text{C}.$$

13. Common Mistakes to Avoid

• Forgetting to convert minutes or hours to seconds before using $Q = I t$.
• Mixing up symbols: $I$ = current, $Q$ = charge, $t$ = time.
• Assuming a constant current when the problem states a varying current – use integration.
• Applying the wrong prefix (e.g., treating 5 mA as 5 A).
• Neglecting the quantisation of charge when asked for the number of electrons transferred.
• Ignoring the internal resistance of an ammeter, which can alter the current being measured.

14. Practice Questions (with Answers)

1. Question: A current of $0.75\;\text{A}$ flows for $2\;\text{h}$. Calculate the charge transferred in coulombs and in mill‑coulombs.

   Answer: $t = 2\;\text{h}=7200\;\text{s}$. $Q = I t = 0.75\times7200 = 5400\;\text{C}=5.4\times10^{6}\;\text{mC}$.

2. Question: A device requires $1.2\;\text{C}$ of charge to operate. If it draws a current of $0.30\;\text{A}$, how long must it be switched on?

   Answer: $t = Q/I = 1.2/0.30 = 4.0\;\text{s}$.

3. Question: A fuse is rated at $5\;\text{A}$. A short‑circuit causes a current of $25\;\text{A}$ for $0.20\;\text{s}$. How much charge passes through the fuse? Will it blow if it melts after $10\;\text{C}$ of charge?

   Answer: $Q = I t = 25\times0.20 = 5.0\;\text{C}$. Since $5.0\;\text{C}<10\;\text{C}$, the fuse will **not** melt during this interval.

4. Question: How many electrons pass a point in a circuit when $2.0\;\text{C}$ of charge flows?

   Answer: $N = Q/e = 2.0/(1.602\times10^{-19}) \approx 1.25\times10^{19}$ electrons.

5. Question: In a copper wire ($n = 8.5\times10^{28}\;\text{m}^{-3}$, $A = 0.5\;\text{mm}^2$) a current of $3\;\text{A}$ flows. Find the drift velocity.

   Answer: $A = 0.5\times10^{-6}\;\text{m}^2$. $v_{d}= \dfrac{3}{(8.5\times10^{28})(0.5\times10^{-6})(1.602\times10^{-19})}\approx4.4\times10^{-4}\;\text{m s}^{-1}$.

6. Question: A current varies as $I(t)=4\sin(\pi t)$ A for $0\le t\le2\;$s. Determine the total charge transferred.

   Answer: $Q=\int_{0}^{2}4\sin(\pi t)\,dt = \frac{4}{\pi}\bigl[1-\cos(2\pi)\bigr]=\frac{8}{\pi}\;\text{C}\approx2.55\;\text{C}$.

15. Linking to Other Syllabus Topics

The concepts mastered here are the foundation for:

• Topic 9.2 – Potential difference: using $V = IR$ (Ohm’s law) together with $Q = I t$ to find energy $W = VQ$.
• Topic 9.3 – Resistance: understanding how $R$ influences $I$ for a given $V$.
• Topic 10 – DC circuits: applying Kirchhoff’s current law (sum of currents at a junction = 0) which directly uses the definition of current.
• Topic 12 – Power and energy: $P = IV = I^{2}R = V^{2}/R$ and $E = Pt = VQt$.

When you move on, keep the quick‑reference box handy – it will appear repeatedly in exam questions.

16. Suggested Diagram (simple series circuit)