To show that the efficiency of power transfer is 50% when maximum power transfer occurs, we need to delve into the concept of maximum power transfer and the relationship between the power delivered to the load and the power supplied by the source.

Step 1: Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that the maximum amount of power is transferred to the load when the load resistance ($R_L$) is equal to the source resistance ($R_S$).

That is: $R_L = R_S$

Step 2: Power Delivered to the Load

Consider a circuit with a voltage source $V$ and a series combination of source resistance $R_S$ and load resistance $R_L$. The total resistance in the circuit is $R_S + R_L$.

The total current ($I$) in the circuit is given by Ohm's law: $I = \frac{V}{R_S + R_L}$

The power delivered to the load $P_L$ is: $P_L = I^2 R_L$

Substituting the expression for $I$: $P_L = \left( \frac{V}{R_S + R_L} \right)^2 R_L$

Step 3: Maximizing Power Delivered to the Load

To maximize the power delivered to the load, we differentiate $P_L$ with respect to $R_L$ and set the derivative equal to zero.

$\frac{dP_L}{dR_L} = \frac{d}{dR_L} \left( \frac{V^2 R_L}{(R_S + R_L)^2} \right)$

This can be simplified, and when we set $\frac{dP_L}{dR_L} = 0$, we find that the power is maximized when: $R_L = R_S$

Step 4: Efficiency of Power Transfer

The efficiency of power transfer $\eta$ is defined as the ratio of the power delivered to the load to the total power supplied by the source.

The total power supplied by the source is: $P_{\text{total}} = I^2 R_S = \left( \frac{V}{R_S + R_L} \right)^2 R_S$

When $R_L = R_S$, we substitute this into the equation for $P_{\text{total}}$: $P_{\text{total}} = \left( \frac{V}{2R_S} \right)^2 R_S = \frac{V^2}{4R_S}$

Now, the efficiency $\eta$ is: $\eta = \frac{P_L}{P_{\text{total}}} = \frac{\frac{V^2}{4R_S}}{\frac{V^2}{4R_S}} = \frac{1}{2} = 50\%$

Conclusion:

Thus, the efficiency of power transfer is 50% when maximum power transfer occurs, which happens when $R_L = R_S$.

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Would you like more details or have any questions?

Here are some related questions you might find interesting:

1. What happens to the efficiency if the load resistance is less than the source resistance?
2. How can the maximum power transfer theorem be applied in electrical engineering design?
3. What is the effect of varying the source voltage on power transfer efficiency?
4. How does the efficiency change if $R_L$ is greater than $R_S$?
5. How can the efficiency be improved in practical circuits?

Tip: The Maximum Power Transfer Theorem is most useful when you want to maximize the power delivered to a load, but for energy efficiency, it's often better to design the circuit such that $R_L$ is larger than $R_S$, to minimize losses.