Question

A cell of internal resistance r drives current through an external resistance R. The power delivered by the cell to the external resistance will be maximum when:
1. R = 0.001 r
2. R = 1000 r
3. R = 2r
4. R = r

Answer 1

Correct Answer - Option 4 : R = r

Concept:

The Maximum Power Transfer Theorem is another useful circuit analysis method to ensure that the maximum amount of power will be dissipated in the load resistance when the value of the load resistance is exactly equal to the resistance of the power source.

The relationship between the load impedance and the internal impedance of the energy source will give the power in the load.

Thus, the power delivered by the cell to the external resistance will be maximum is when R = r

Current, \(i = \frac{E}{{r + R}}\)

Power generated in R, P = i²R

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

Calculation:

For maximum power \(\;\frac{{dP}}{{dR}} = 0\)

\({E_2}\left[ {\frac{{{{(r + R)}^2} \times 1 - R \times 2\left( {r + R} \right)}}{{{{(r + R)}^4}}}} \right] = 0\) 

Therefore R = r