Question

When a resistor of R ohms is connected across a battery with electromotive force E volts and internal resistance r ohms, a current of I amperes will flow, generating P watts of power, where I=E/r+R and P=I²R

Assuming r is constant, what choice of R results in maximum power?

Answer 1

The maximum power transfer occurs when the load resistance R is equal to the internal resistance r of the battery. This is derived by substituting I = E / (r + R) into the power equation and using calculus to find the maximum value of the function.

Step-by-step explanation:

To determine the choice of R that results in maximum power dissipation, we need to look at the power equation P = I²R, where I is the current and R is the resistance of the load resistor. Given that I = E / (r + R), if we substitute for I in the power equation, we get P = (E² / (r + R)²) * R. To maximize P with respect to R, we need to take the derivative of P with respect to R and set it to zero, then solve for R. This optimization problem is an application of calculus and is commonly known as the 'Maximum Power Transfer Theorem,' which states that the power is maximized when the load resistance R is equal to the internal resistance r. Therefore, the choice of R that results in maximum power is R = r.