Show that the power dissipated by the load resistor is maximum when the resistance of the load r

Show that the power dissipated by the load resistor is...

Key Concepts

Optimization in Circuit Analysis

Circuit analysis often involves optimizing certain parameters, such as maximizing power transfer. This is typically achieved by expressing the power dissipated in the load as a function of load resistance and then using differentiation to find its maximum value, highlighting the interplay between mathematical techniques and electrical engineering principles.

Load Resistance and Power Dissipation

The load resistance is the resistance in the circuit that consumes the power supplied by the source. The power dissipated in a resistive element is expressed as a function of voltage and resistance, and understanding this relationship is key to determining conditions—such as in the Maximum Power Transfer Theorem—under which the load receives the highest possible power.

Internal Resistance

Internal resistance represents the inherent resistance within a power source such as a battery. It causes a voltage drop within the source itself and plays a crucial role in the operation of practical circuits by affecting how much of the source’s voltage is available to drive current through an external load.

Maximum Power Transfer Theorem

This theorem states that the power delivered to a load in an electrical circuit is maximized when the load resistance is equal to the source's internal (or Thevenin) resistance. It is a fundamental concept in circuit design, ensuring that energy is utilized most efficiently when transferring power from a source to a load.

Transcript

00:01 In this question, i want to prove that the power dissipated by our load resistor is maximum.

00:10 When our external load resistance is equals to our internal resistance.

00:19 Now let us draw the diagram for you.

00:23 This is our battery with some voltage across it.

00:28 And we have the internal resistance.

00:31 R is connected to an external load resistor resistance r.

00:39 Now to find what is the power dissipated by our load resistor, we use the equation.

00:49 Power is equal to i square r.

00:53 Since we will be quite simpler to find what is the total current rather than the potential difference across the or r.

01:06 So in this case the current is governed by the total potential difference v divided by the total resistance of this circuit which is r plus big r.

01:27 From here what we need to do is to differentiate to find what is the change in the power deservators with respect to big r right and to investigate how the power the speed of change when the pr changes.

01:43 So in this case, we'll be taking e .p.

01:54 And what we actually want is for the rate of change of this power to be 0, to find what is the maximum point, right? so this is equals to taking the differentiation of the entire expression.

02:23 Plus r square.

02:28 Now take note that v square of v is a constant in this case.

02:33 Since we are not investigating how the voltage changes, so we can divide v to the left hand side, which is zero, which is getting rid of v square...

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