If the power in the external resistance R is | vedclass

Name: Exam Paper Generator | JEE NEET Paper Generator | Vedclass
Brand: Vedclass
Rating: 5 (35 reviews)

Question

(D) For a circuit with an electromotive force $E$ and internal resistance $r$,the power $P$ delivered to an external resistance $R$ is given by $P = I

Vedclass · Accepted Answer

All

Vedclass · Answer

(D) For a circuit with an electromotive force $E$ and internal resistance $r$,the power $P$ delivered to an external resistance $R$ is given by $P = I^2 R = \left(\frac{E}{R+r}ight)^2 R$.To find the condition for maximum power,we differentiate $P$ with respect to $R$ and set it to zero: $\frac{dP}{dR} = E^2 \left[ \frac{(R+r)^2 - R \cdot 2(R+r)}{(R+r)^4} ight] = 0$.This gives $R+r - 2R = 0$,so $R = r$. Thus,statement $(i)$ is correct.Substituting $R = r$ into the power formula: $P = \left(\frac{E}{R+R}ight)^2 R = \left(\frac{E}{2R}ight)^2 R = \frac{E^2}{4R^2} \cdot R = \frac{E^2}{4R}$. Thus,statement $(ii)$ is correct.The total power supplied by the source (input power) is $P_{	ext{in}} = E \cdot I = E \cdot \left(\frac{E}{R+r}ight)$. Since $R=r$,$P_{	ext{in}} = E \cdot \left(\frac{E}{2R}ight) = \frac{E^2}{2R}$. Thus,statement $(iii)$ is correct.The efficiency $\eta$ is defined as the ratio of output power to input power: $\eta = \frac{P_{	ext{out}}}{P_{	ext{in}}} 	imes 100 = \frac{E^2/4R}{E^2/2R} 	imes 100 = \frac{1}{2} 	imes 100 = 50\%$. Thus,statement $(iv)$ is correct.Since all statements are correct,the correct option is $D$.

If the power in the external resistance $R$ is maximum,then which of the following statements are correct?
$(i) R = r$
$(ii)$ Power in $R$ is $\frac{E^2}{4R}$
$(iii)$ Input power is $\frac{E^2}{2R}$
$(iv)$ Efficiency is $50\%$

Similar Questions

$A$ cable of resistance $10\,\Omega$ carries electric power from a generator producing $250\,kW$ at $10000\,V$. The power lost in the cable during transmission is ............. $kW$.

How many $60 \,W$ bulbs may be safely run on $220 \,V$ using a $5 \,A$ fuse?

$A$ current of $2\, A$ passing through a conductor produces $80\, J$ of heat in $10$ seconds. The resistance of the conductor is ............ $\Omega$.

The ratio of heat dissipated per second through the resistance $5 \Omega$ and $10 \Omega$ in the circuit given below is:

There are $n$ number of identical electric bulbs,each designed to draw a power $P$ independently from the mains supply. If they are now joined in series across the same mains supply,what is the total power drawn by the combination?

Vedclass Test Series

Exam Paper Generator

Online Exam Module

If the power in the external resistance $R$ is maximum,then which of the following statements are correct?$(i) R = r$$(ii)$ Power in $R$ is $\frac{E^2}{4R}$$(iii)$ Input power is $\frac{E^2}{2R}$$(iv)$ Efficiency is $50\%$