Ch 28: Fundamentals of Circuits

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 28: Fundamentals of Circuits

Problem 51a

Chapter 28, Problem 51a

Load resistor R is attached to a battery of emf and internal resistance r. For what value of the resistance R, in terms of ∈ and r, will the power dissipated by the load resistor be a maximum?

Verified step by step guidance

Step 1: Start by recalling the formula for power dissipated in a resistor. The power dissipated by the load resistor R is given by \( P = I^2 R \), where \( I \) is the current through the circuit.

Step 2: Use Ohm's Law to express the current \( I \) in terms of the emf \( \varepsilon \), the internal resistance \( r \), and the load resistance \( R \). The total resistance in the circuit is \( R + r \), so \( I = \frac{\varepsilon}{R + r} \). Substitute this expression for \( I \) into the power formula.

Step 3: Substitute \( I = \frac{\varepsilon}{R + r} \) into \( P = I^2 R \). This gives \( P = \frac{\varepsilon^2 R}{(R + r)^2} \). This is the expression for power dissipated in the load resistor.

Step 4: To find the value of \( R \) that maximizes \( P \), take the derivative of \( P \) with respect to \( R \) and set it equal to zero. Use the quotient rule for differentiation: \( \frac{d}{dR} \left( \frac{\varepsilon^2 R}{(R + r)^2} \right) = 0 \). Solve this equation for \( R \).

Step 5: After solving the derivative equation, you will find that the power is maximized when \( R = r \). This result is known as the condition for maximum power transfer, which states that the load resistance should equal the internal resistance of the battery for maximum power dissipation.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Ohm's Law

Ohm's Law states that the current (I) flowing through a conductor between two points is directly proportional to the voltage (V) across the two points and inversely proportional to the resistance (R) of the conductor. This relationship is expressed as V = IR. Understanding this law is crucial for analyzing circuits, as it helps determine how voltage, current, and resistance interact.

Power in Electrical Circuits

The power (P) dissipated in a resistor is given by the formula P = I²R, where I is the current through the resistor and R is its resistance. In the context of load resistors, maximizing power dissipation involves finding the optimal resistance value that allows for the most efficient conversion of electrical energy into heat or work, which is essential for circuit design.

Maximum Power Transfer Theorem

The Maximum Power Transfer Theorem states that maximum power is delivered to a load when the load resistance (R) is equal to the internal resistance (r) of the source. This principle is vital for optimizing circuit performance, as it guides the selection of load resistors to ensure that the maximum possible power is transferred from the source to the load.

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