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Electric circuits: Ohm’s law, resistors and the cost of electricity

الأسبوع 8 · الدرس الثاني «الدوائر الكهربية»: قانون أوم وتوصيل المقاومات على التوالي وعلى التوازي والقدرة والطاقة الكهربية وتكلفة الاستهلاك المنزلي والسلامة · 3 دروس
1

Ohm's law: V = I × R

Science Prep 3 Term 1, Unit 2: pp. 46–53
GoalUse Ohm’s law, I = V/R, to find a current, a voltage or a resistance, and read the slope of an I–V graph.
New words
Ohm’s law · at a constant temperature, the current in a fixed resistor is directly proportional to the voltage across it: I = V/Rdirectly proportional · when one quantity doubles, the other doubles tooslope · how steep a graph line is: the change in the vertical quantity divided by the change in the horizontal quantity
Explanation

The German physicist Georg Ohm (1789–1854) found a rule for resistors at a steady temperature: double the voltage across a fixed resistor and the current doubles too. So I is directly proportional to V, and I = V ÷ R, or V = I × R. The units are volts (V), amperes (A) and ohms (Ω), so 1 Ω = 1 V/A. If we draw the current I against the voltage V for a resistor, we get a straight line through the origin. Its slope is 1/R, so R = change in V ÷ change in I. Careful: Ohm’s law holds for a conductor at a constant temperature. A bulb filament, which heats up, does not give a straight line, so we must not claim it does. To solve a problem, write the formula, put in the numbers with their units, and calculate. If the resistance is the unknown, use R = V ÷ I; if the voltage is unknown, use V = I × R. Never write R = I ÷ V, and never say that a resistance changes because the voltage changes.

V (volts)I (amperes)2468100.511.520
Worked examples
A 9 V cell is joined to a 3 Ω resistor. I = V ÷ R = 9 V ÷ 3 Ω = 3 A.
A current of 4 A flows when 20 V is applied. R = V ÷ I = 20 V ÷ 4 A = 5 Ω.
A current of 0.5 A flows through a 24 Ω resistor. V = I × R = 0.5 A × 24 Ω = 12 V.
V (V)I (A)V ÷ I (Ω)
20.45
40.85
61.25
81.65
102.05
Invented readings for one resistor: V = 2, 4, 6, 8, 10 V give I = 0.4, 0.8, 1.2, 1.6, 2.0 A. The ratio V ÷ I is 5 Ω every time, so this is an ohmic resistor of 5 Ω. From the graph, slope = 0.8 A ÷ 4 V = 0.2 A/V, and 1 ÷ 0.2 = 5 Ω.
Class activity

Graph from readings, with a grown-up. Draw I–V axes on squared paper. Plot the six points V = 0, 2, 4, 6, 8, 10 V and I = 0, 0.4, 0.8, 1.2, 1.6, 2.0 A. Draw the straight line through them. Find the slope between two points and then 1 ÷ slope: that is R. Now invent your own readings for a 10 Ω resistor, plot them and compare the two lines. No real circuit is needed.

Practice
1
Find I when 15 V is applied across 5 Ω.
2
A resistor passes 2 A when the voltage across it is 14 V. Find R.
3
What is the voltage across a 6 Ω resistor that carries 1.5 A?
4
The voltage across a resistor is doubled, from 3 V to 6 V. What happens to the current, if the temperature is constant?
5
Why can we not use Ohm’s law as a straight line for a bulb filament?