Question 120: Current Electricity

Question

An electric power line having total resistance of $2 \Omega$, delivers 1 kW of power at 250 V . The percentage efficiency of transmission line is $\_\_\_\_$ .

Accepted Answer

The resistance of the transmission line, $\mathrm{R}=2 \Omega$
The power delivered (Output Power), $\mathrm{P}_{	ext {out }}=1 \mathrm{~kW}=1000 \mathrm{~W}$
The given power is delivered at voltage, $\mathrm{V}=250 \mathrm{~V}$
The power delivered to the load is the product of the voltage across the load and the current flowing through it.
$$ \mathrm{P}_{\mathrm{out}}=\mathrm{V} \cdot \mathrm{I} \Rightarrow \mathrm{I}=\frac{\mathrm{P}_{\mathrm{out}}}{\mathrm{~V}} $$
$$ \Rightarrow I=\frac{1000}{250} A=4 A $$
So, a current of 4 A flows through the transmission line.
As the current flows through the transmission wires, some electrical energy is lost as heat due to the line's resistance. This power loss ( $\mathrm{P}_{	ext {loss }}$ ) is calculated using Joule's heating formula :
$$ \mathrm{P}_{	ext {loss }}=\mathrm{I}^2 \cdot \mathrm{R} $$
$$\Rightarrow $$ $$ \mathrm{P}_{\mathrm{loss}}=(4)^2 	imes 2 \mathrm{~W}=32 \mathrm{~W} $$
The total power supplied at the source (Input Power, $\mathrm{P}_{\mathrm{in}}$ ) must equal the power successfully delivered to the load plus the power lost in the transmission line.
$$ \mathrm{P}_{\mathrm{in}}=\mathrm{P}_{\mathrm{out}}+\mathrm{P}_{	ext {loss }} $$
$$\Rightarrow $$ $\mathrm{P}_{\mathrm{in}}=1000+32$
$$\Rightarrow $$ $$ \mathrm{P}_{\mathrm{in}}=1032 \mathrm{~W} $$
The efficiency $(\eta)$ of the transmission line is the ratio of the useful power delivered to the total power supplied, expressed as a percentage :
$$ \eta=\left(\frac{P_{	ext {out }}}{P_{	ext {in }}}ight) 	imes 100 \% $$
$$\Rightarrow $$ $$ \eta=\left(\frac{1000}{1032}ight) 	imes 100 \% \approx 96.9 \% $$
Therefore, the percentage efficiency of the transmission line is $96.9 \%$. Hence, the correct option is (A).

Answer

96.9

Answer

92.5

Answer

86.5

Answer

100

Current Electricity

An electric power line having total resistance of 2 \Omega, delivers 1 kW of power at 250 V . The percentage efficiency of transmission line is \_\_\_\_ .

Choose the correct answer: