# Electrical Energy & Power

Electric energy is the amount of energy that is transferred by an electric current over a period of time. It is measured in units of joules (J) or watt-hours (Wh). The energy transferred by a current $I$ to a load resistance $R$ in time $t$ is
\begin{align}
E=I^2Rt.
\end{align}

**Electric power** is the rate at which electric energy is transferred. It is the amount of electric energy that is delivered or consumed per unit time. Electric power is measured in units of watts (W) or kilowatts (kW). The power delivered by a current $I$ to a load resistance $R$ is
\begin{align}
P=I^2R=VI.
\end{align}

The amount of electric energy consumed depends on the power of the device and the amount of time it is used. For example, a 100-watt light bulb that is used for 10 hours consumes 1,000 watt-hours or 1 kilowatt-hour (kWh) of electric energy.

## Problems from IIT JEE

**Problem (IIT JEE 2002):**

**Problem:**
A 100 W bulb B_{1} and two 60 W bulbs B_{2} and B_{3}, are connected to a 250 V source, as shown in the figure. Now W_{1}, W_{2} and W_{3} are the output powers of the bulb B_{1}, B_{2} and B_{3} respectively. Then,

- $W_1 > W_2 = W_3$
- $W_1 > W_2 > W_3$
- $W_1 < W_2 = W_3$
- $W_1 < W_2 < W_3$

**Solution:**
Let the given power ratings be defined at an operating voltage V. The resistances of the three bulbs are given by
\begin{align}
R_1&={V^2}/{100},\\
R_2&={V^2}/{60},\\
R_3&={V^2}/{60}.\nonumber
\end{align}
In the given configuration, the current through $B_1$ and $B_2$ is
\begin{alignat}{2}
i&=\frac{250}{R_1+R_2} \\
&=\frac{250}{V^2}\left(\frac{100\times 60}{100+60}\right) \\
&=\frac{250}{V^2}\left(\frac{75}{2}\right).\nonumber
\end{alignat}
Substitute the values of $i$, $R_1$, and $R_2$ to get the output powers
\begin{align}
&W_1=i^2R_1 \approx 14\, \left({250}/{V}\right)^{2}, \nonumber\\
&W_2=i^2R_2 \approx 23\, \left({250}/{V}\right)^{2}, \nonumber
\end{align}
and
\begin{align}
&W_3={(250)^2}/{R_3}=60\, \left({250}/{V}\right)^{2}.\nonumber
\end{align}
It is interesting to note that $B_1$ (100 W) is dimmer than $B_2$ (60 W) which in turn is dimmer than $B_3$ (60 W).

**Problem (JEE Mains 2022):**
An electric bulb rated (220 V, 100 W) is connected in series with another bulb rated (220 V, 60 W). If the voltage across combination is 220 V, the power consumed by the 100 W bulb will be about______Watt.

## Related

- Ohm's Law
- Electrical Resistance
- Electric Cell
- Kirchhoff's Laws
- Heating Effect of Current
- Thermo-electricity