PMDC Unit 45 · 5 outcomes

Current Electricity

Unit 9 is a steady flow of charge, Ohm’s law, how resistivity cares about temperature, the hidden r inside a battery, and when a load takes maximum power.

Open the Unit 45 quiz Worked numericals

9.1

Steady current

PMDC: Describe the concept of steady current.

Current is charge passing a point per second. Steady (direct) current means that rate does not change with time — the same I right now and a second later. A battery plus a resistor, after the brief start-up, is the picture. AC is not steady; it keeps reversing.

Live graph — drag the sliders

I = Q / t = 4/2 = 2 A

$I = Q / t$
$1 A = 1 C s^{- 1}$
steady: I constant in time

Hold these

Conventional current is the flow of + charge: from the + terminal around to the −.
In a metal the moving charges are electrons, opposite to that arrow.

9.2

Ohm’s law

PMDC: State Ohm’s law.

For a metal at fixed temperature, current is proportional to the potential difference. The graph of V against I is a straight line through the origin. The slope is resistance R.

Live graph — drag the sliders

I = V / R = 2

A through origin

$V = I$ R (Ohm, metal, constant T)
$R = V / I$

Hold these

Ohm’s law is not a definition of R. It is the statement that R stays put when V changes.
A filament lamp gets hot; R rises; the V–I graph curves. Not ohmic.

9.3

Resistivity and temperature

PMDC: Define resistivity and explain its dependence upon temperature.

Resistance of a wire depends on material, length and thickness. Resistivity $ρ$ is the material part: $R = ρ$ L / A. Long thin wire, large R.

For metals, hotter means more vibration, more scattering, higher $ρ$ . For semiconductors the opposite: hotter frees more carriers, $ρ$ falls. The temperature coefficient $α$ tells you the fractional rise per kelvin.

Live graph — drag the sliders

L \times 2 A \div 2 R \to 4 R_{0}

$R = ρ L / A$
$ρ = R A / L u ni t : Ω m$
$ρ_{T} = ρ_{0} [1 + α (T - T_{0})]$
$R_{T} = R_{0} [1 + α (T - T_{0})]$

Hold these

Metals: $α$ > 0. Carbon / semiconductors: $α$ often negative.
A is cross-section, not surface area.

9.4

Internal resistance

PMDC: Explain the internal resistance of sources and its consequences for external circuits.

A real cell is an emf E in series with a small internal resistance r. When current I flows, some voltage is dropped inside the cell: I r. What you measure across the terminals is less than E.

Consequence: the more current you draw, the more the terminal voltage sags. A flat battery has a large r.

Live graph — drag the sliders

V = E - I r = 12 - 4 = 8 V

$V = E$ − I r (terminal voltage, discharging)
$E = I (R + r)$
$I = E / (R + r)$

Hold these

Open circuit ( $I = 0) : V = E .$
Short circuit (R → 0): $I = E$ / r (large, and V → 0).

9.5

Maximum power transfer

PMDC: Describe the conditions for maximum power transfer.

Power in the load is $P = I^{2}$ R. I itself depends on R: $I = E$ / (R + r). P is small if R is tiny (almost a short, but V is tiny) and small if R is huge (almost no current). The peak sits at $R = r .$

That is the matching condition. Efficiency is only 50 % there — half the power is cooked in r — so power lines are not matched this way. The rule is for getting the most out of a given cell into a load you may choose.

Live graph — drag the sliders

R = 3Ω r = 3Ω P = 12 W

$P = I^{2} R = E^{2} R / (R + r)^{2}$
$P_{m a x}$ when $R = r$
$P_{m a x} = E^{2} / (4 r)$

Hold these

Maximum power ≠ maximum efficiency.
Match R to r for max P. For less waste, use R ≫ r.

Unit 45 formula sheet

Write these until they are cheaper than thinking, then open the quiz.

$I = Q / t$
$1 A = 1 C s^{- 1}$
steady: I constant in time
$V = I$ R (Ohm, metal, constant T)
$R = V / I$
$R = ρ L / A$
$ρ = R A / L u ni t : Ω m$
$ρ_{T} = ρ_{0} [1 + α (T - T_{0})]$
$R_{T} = R_{0} [1 + α (T - T_{0})]$
$V = E$ − I r (terminal voltage, discharging)
$E = I (R + r)$
$I = E / (R + r)$
$P = I^{2} R = E^{2} R / (R + r)^{2}$
$P_{m a x}$ when $R = r$
$P_{m a x} = E^{2} / (4 r)$

Open the Unit 45 quiz