Calculate electrical power (P = VI, P = I²R)

WHAT is electrical power?

WHY do we care (Hardware angle): Every component has a power rating. A resistor rated at 0.25 W will literally burn if you push more power through it. Power = heat = the thing that melts your PSU, throttles your CPU, and drains your battery. Knowing $P$ tells you heat and energy use.

HOW to derive $P = VI$ from first principles

Start from the definitions — don't memorize, build it.

Voltage = energy per unit charge: $V = \dfrac{E}{Q}\;\Rightarrow\; E = VQ$ . Why this step? Voltage literally means "joules given to each coulomb," so total energy is voltage times how much charge you moved.
Current = charge per unit time: $I = \dfrac{Q}{t}\;\Rightarrow\; Q = It$ . Why this step? Current tells us how fast charge flows, so charge = current × time.
Substitute into power $P = \dfrac{E}{t}$ : $P = \frac{E}{t} = \frac{VQ}{t} = \frac{V(It)}{t} = VI$ Why this step? The $t$ cancels — energy per second becomes voltage × current directly.

HOW to get $P = I^2R$ and $P = V^2/R$

We already have $P = VI$ . Now bring in Ohm's law: $V = IR$ .

Replace $V$ : $P = VI = (IR)\,I = I^2R$

Replace $I$ (since $I = V/R$ ): $P = VI = V\cdot\frac{V}{R} = \frac{V^2}{R}$

Figure — Calculate electrical power (P = VI, P = I²R)

Worked examples

Common mistakes (Steel-man + fix)

Recall Feynman: explain to a 12-year-old

Imagine a water slide. Voltage is how tall the slide is (how much of a drop). Current is how many kids slide down each second. Power is how much total whee! energy happens per second — a tall slide with lots of kids gives loads of power. If the slide is rough (resistance), the kids rub against it and it gets hot — that heat is $I^2R$ , and it grows really fast if more kids pile on, because they slow each other and press harder.

Flashcards

What is the SI unit of power and what does it equal?

The watt (W);

1\ \text{W} = 1\ \text{J/s}

Derive

P=VI

from definitions.

V=E/Q

and

I=Q/t

;

P=E/t = VQ/t = V(It)/t = VI

How do you get

P=I^2R

from

P=VI

Substitute Ohm's law

V=IR

P=(IR)I=I^2R

How do you get

P=V^2/R

Substitute

I=V/R

into

P=VI

P=V(V/R)=V^2/R

Which formula if you know current and resistance but not voltage?

P=I^2R

Which formula if you know voltage and resistance but not current?

P=V^2/R

If current in a fixed resistor doubles, how does power change?

It quadruples (×4), because

P\propto I^2

A 5 V, 2 A charger delivers how much power?

P=VI=10\ \text{W}

Energy from a 60 W device in 2 hours (joules)?

E=Pt=60\times7200=432{,}000\ \text{J}

Why does thick, low-resistance wire reduce heat loss?

Lower

R

means lower

I^2R

heat for the same current.

Connections

Ohm's Law (V = IR) — the bridge that turns $P=VI$ into $I^2R$ and $V^2/R$ .
Voltage and Potential Difference — energy per charge, the source of the $VQ$ term.
Electric Current — charge per time, the source of the $It$ term.
Energy and the Joule — $E = Pt$ , billing (kWh) and battery capacity.
Resistors and Power Ratings — why a 0.25 W resistor burns out.
Heat Dissipation and Cooling — $I^2R$ losses in CPUs, wires, PSUs.

Concept Map

Hinglish (regional understanding)

Intuition Hinglish mein samjho

Dekho, power ka matlab hai energy kitni tezi se transfer ho rahi hai — units mein watts, matlab joules per second. Circuit mein charge jab ek voltage difference ke through move karta hai, toh woh kaam karta hai (energy deta hai). Isiliye base formula hai $P = VI$ : voltage batata hai har coulomb ko kitni energy mili ( $V=E/Q$ ), aur current batata hai per second kitna charge beh raha hai ( $I=Q/t$ ). Dono multiply karo, time cancel ho jaata hai, aur mil jaata hai power.

Ab Ohm's law $V=IR$ ka use karke hum wahi formula do aur tarike se likh sakte hai: $P=I^2R$ aur $P=V^2/R$ . Yeh teeno same cheez hai, bas alag quantities ke liye convenient. Jo pata ho use karo: V aur I pata hai toh $VI$ ; I aur R pata hai toh $I^2R$ ; V aur R pata hai toh $V^2/R$ .

Sabse important insight: $I^2R$ mein current ka square hai. Matlab current double karo toh heat char guna ho jaati hai, do guna nahi! Isiliye hardware mein thick (low resistance) wires use karte hai aur high voltage se power bhejte hai — taaki current kam rahe aur $I^2R$ heat loss kam ho. Yeh reason hai ki CPU, PSU, aur battery sab power (heat) ke around design hote hai. Ek aur common galti: time ko seconds mein convert karna mat bhoolna jab energy joules mein chahiye (2 hours = 7200 s).

Calculate electrical power (P = VI, P = I²R)

WHAT is electrical power?

HOW to derive $P = VI$ from first principles

HOW to get $P = I^2R$ and $P = V^2/R$

Worked examples

Common mistakes (Steel-man + fix)

Flashcards

Connections

Concept Map

Hinglish (regional understanding)

Go deeper — visual, from zero

Test yourself — Electricity & Charge Basics

Connections