Why Does External Resistance Equal Internal Resistance for Maximum Power?

[sArGe99]

Homework Statement

My textbook says that the power delivered by the battery to external resistance network will be maximum when the external resistance equals the internal resistance? Why is it so?
p = I^2 * R, so greater the external resistance, greater the power(even if it exceeds internal resistance)

Homework Equations

P = VI ;)

The Attempt at a Solution

Clueless about this one.

 

[rl.bhat]

In the circuit the current is written as
I = E/(R + r) where E is the emf of the battery and r is the internal resistance of the battery.
So P = I^2*R =[ E^2/(R + r)^2]*R
The power delivered by the battery depends on the external resistance. So find
dP/dR and equate it to zero to get the condition for maximum power delivery

 

[sArGe99]

Thanks a lot.