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.
- Convert time: \$t = 3.5\;\text{min}\times60\;\frac{\text{s}}{\text{min}} = 210\;\text{s}\$.
- 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}\$.
- Convert area: \$A = 1.0\;\text{mm}^2 = 1.0\times10^{-6}\;\text{m}^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)
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}\$.
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}\$.
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.
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.
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}\$.
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)
