Units, Prefixes, Ratios and Straight-line Graphs
This chapter has three short formulas – V = W/Q, I = Q/t and V = I R – and three new units: the volt, the ampere and the ohm. Questions mix in milliamperes, kilohms and microamperes, ask you to spot a constant ratio in a table, to read the slope of a straight-line graph, and to add fractions such as 1/6 + 1/3 for resistors in parallel. This page practises each of these with numbers before the physics arrives.
- Multiplying and dividing by 10, 100 and 1000.
- Simple fractions (½, ⅓, ¼).
1. The units of this chapter
Every quantity has a symbol (a letter used in formulas) and a unit. The three new units are all built from units you already know – the coulomb, the joule and the second:
| Quantity | Symbol | Unit | Built from |
|---|---|---|---|
| charge | Q | coulomb (C) കൂളോം | – (page 1) |
| time | t | second (s) | – |
| work, energy | W | joule (J) | 1 J = 1 N × 1 m |
| potential difference | V | volt (V) വോൾട്ട് | 1 V = 1 J/C (V = W/Q) |
| current | I | ampere (A) ആമ്പിയർ | 1 A = 1 C/s (I = Q/t) |
| resistance | R | ohm (Ω) ഓം | 1 Ω = 1 V/A (R = V/I) |
The letter V is both the symbol for potential difference and the unit volt: “V = 6 V” means “the potential difference is 6 volts”. Similarly W stands for work in this chapter (in J), not for the watt. Always write a space between a number and its unit: 1.5 V, 0.5 A, 10 Ω.
Look at the bottom line: the units are multiplied or divided exactly like the numbers. If your units come out wrong, your formula is wrong.
2. Prefixes: milli, micro, kilo and mega
Real currents and resistances can be very small or very large, so we put a prefix in front of the unit:
| Prefix | Symbol | Means | Power of ten | Example |
|---|---|---|---|---|
| mega | M | × 1 000 000 | 106 | 1 MΩ = 1 000 000 Ω = 1000 kΩ |
| kilo | k | × 1000 | 103 | 1 kΩ = 1000 Ω |
| milli | m | ÷ 1000 | 10−3 | 1 mA = 0.001 A, so 1 A = 1000 mA |
| micro | µ | ÷ 1 000 000 | 10−6 | 1 µA = 0.000 001 A, so 1 mA = 1000 µA |
A power of ten just counts zeros: 103 = 1000 (three zeros), and 10−3 = 1 ÷ 1000 = 0.001. Multiplying by 103 moves the decimal point 3 places right; multiplying by 10−3 moves it 3 places left. Your textbook (page 140) uses milliampere (മില്ലി ആമ്പിയർ) and microampere (മൈക്രോ ആമ്പിയർ); page 145 uses kΩ and MΩ.
Before using any formula, change every value to the base unit (A, V, Ω, C). A current of 250 mA must go into V = I R as 0.25 A.
Worked example 1 · Milliamperes to amperes
Q. A torch bulb draws a current of 250 mA. Write this current in amperes.
1 A = 1000 mA, so to go from mA to A, divide by 1000 (move the point 3 places left).
Worked example 2 · Kilohms to ohms
Q. A resistor is marked 4.7 kΩ. What is its resistance in ohms? Write it also as a power of ten.
1 kΩ = 1000 Ω = 103 Ω, so multiply by 1000 (move the point 3 places right).
3. Direct proportion and a constant ratio
Two quantities are directly proportional when doubling one doubles the other, tripling one triples the other, and zero of one gives zero of the other. We write y ∝ x. The test is simple: the ratio y ÷ x stays the same in every row of the table. That constant ratio is often a useful quantity in its own right. Example: torch cells joined one after another (in series, page 6) – each 1.5 V cell adds 1.5 V, so the voltage is proportional to the number of cells and V ÷ n = 1.5 V every time.
Proportional data gives a constant ratio and a straight line through the origin (0, 0). The square example gives a curve, and the ratio keeps changing – so area is not proportional to the side.
In Ohm's law (page 8) the ratio V ÷ I is constant for a given wire, and that constant is the resistance R.
Inverse proportion – the opposite pattern
Sometimes when one quantity doubles, the other halves. Then their product stays the same, and we say they are inversely proportional. With a fixed battery, the current is inversely proportional to the resistance: I = V ÷ R. More resistance → less current.
Try R = 3 Ω and then R = 6 Ω: the resistance doubles and the current halves, from 4 A to 2 A. The product I × R is always 12 – the battery's voltage.
4. Straight-line graphs through the origin – and their slope
To draw a graph from a table, choose a scale for each axis, plot each pair as a point, and join them. If the points lie on a straight line through the origin, the two quantities are directly proportional. The slope (steepness) of the line is
For a straight line through the origin, the slope is the same as the constant ratio y ÷ x for any point on it. In this chapter, potential difference V is plotted on the y-axis and current I on the x-axis (Graph 7.1), so the slope V ÷ I is the resistance. A steeper line means a larger resistance.
Wherever you put the triangle, rise ÷ run gives the same number – that is what makes the line straight. Compare slopes 3, 6 and 9: the steeper line belongs to the bigger resistance.
Worked example 3 · Slope from a graph
Q. A straight-line V–I graph through the origin passes through the points (0.2 A, 2 V) and (0.4 A, 4 V). Find its slope. What does the slope tell you?
Rise = change in V between the two points; run = change in I. Slope = rise ÷ run, in V/A, which is the ohm.
5. Adding fractions such as 1/R = 1/6 + 1/3
For resistors in parallel (page 10) you will add reciprocals (“one over” a number) and then turn the answer upside down. The steps are always the same: (1) write both fractions with a common denominator, (2) add the tops, (3) simplify, (4) flip the fraction to get R. Do not forget step 4 – the question asks for R, not 1/R!
The strip is one whole. The blue part is 1/R₁ of it and the green part 1/R₂. Together they make 1/R. A shortcut for two values: R = (R₁ × R₂) ÷ (R₁ + R₂) – “product over sum”. Check that it gives the same answer. Notice that R is always smaller than the smaller of R₁ and R₂.
Worked example 4 · 1/R = 1/6 + 1/3
Q. Find R if 1/R = 1/6 + 1/3.
The common denominator is 6: 1/3 = 2/6. Add, simplify, then flip.
Look at the labels around you: a torch cell says 1.5 V, a power bank may say 10000 mAh (that is 10 A for one hour – the mAh measures charge), a phone charger says 5 V ⎓ 2 A, and the KSEB supply to your house is 230 V. Electronic resistors are often marked in kΩ. Being quick with these prefixes makes the chapter's numericals easy.