A battery of e. m. f. E and internal resistance r is connected to an external resistance R the condition for maximum power transfer is

Question

A battery of e. m. f. E and internal resistance r is connected to an external resistance R the condition for maximum power transfer is (A)  r < R  (B)  r>R  (C)  r=1/R  (D)  r=R

Tardigrade Admin · Accepted Answer

We know that the current in the circuit

I = \frac{E}{R + r}

and power delivered to the resistance

R

is

P = I^{2} R = \frac{E ^{2} R}{( R + r ) ^{2}}

It is maximum when

\frac{d P}{d R} = 0

\frac{d P}{d R} = E^{2} [\frac{( r + R ) ^{2} - 2 R ( r + R )}{( r + R ) ^{4}}] = 0

or

(r + R)^{2} = 2 R (r + R)

or

R = r

Q. A battery of e. m. f. $E$ and internal resistance $r$ is connected to an external resistance $R$ the condition for maximum power transfer is

$r < R$

$r > R$

$r = 1/ R$

$r = R$

Q. A battery of e. m. f. E and internal resistance r is connected to an external resistance R the condition for maximum power transfer is

r<R

r>R

r=1/R

r=R

We know that the current in the circuit I=R+rE​ and power delivered to the resistance R is P=I2R=(R+r)2E2R​ It is maximum when dRdP​=0 dRdP​=E2[(r+R)4(r+R)2−2R(r+R)​]=0 or (r+R)2=2R(r+R) or R=r

Q. A battery of e. m. f. $E$ and internal resistance $r$ is connected to an external resistance $R$ the condition for maximum power transfer is

$r < R$

$r > R$

$r = 1/ R$

$r = R$