Problem 111 - Physics

Question

Interactive Solution \(\underline{20.111}\) at provides one approach to problems like this one. Three resistors are connected in series across a battery. The value of each resistance and its maximum power rating are as follows: \(2.0 \Omega\) and \(4.0 \mathrm{~W}, 12.0 \Omega\) and \(10.0 \mathrm{~W}\), and \(3.0 \Omega\) and \(5.0 \mathrm{~W}\). (a) What is the greatest voltage that the battery can have without one of the resistors burning up? (b) How much power does the battery deliver to the circuit in (a)?

Step-by-Step Solution

Answer

The maximum voltage is 15.47 V, and the power delivered is 14.08 W.

Step 1: Identify the Problem

We need to determine the maximum voltage the battery can have without exceeding the power rating of any resistor in the series combination.

Step 2: Calculate Maximum Current for Each Resistor

Using the power formula, \( P = I^2 R \), where \( P \) is the power rating and \( R \) is the resistance, we find the maximum current for each resistor:- For \( 2.0 \, \Omega \) resistor: \( I_{max} = \sqrt{\frac{4.0}{2.0}} = 1.41 \, \text{A} \).- For \( 12.0 \, \Omega \) resistor: \( I_{max} = \sqrt{\frac{10.0}{12.0}} = 0.91 \, \text{A} \).- For \( 3.0 \, \Omega \) resistor: \( I_{max} = \sqrt{\frac{5.0}{3.0}} = 1.29 \, \text{A} \).

Step 3: Determine Maximum Allowable Current

The current through the series circuit is the same for each resistor. Therefore, the maximum allowable current is determined by the smallest current value calculated in the previous step, \( I_{max} = 0.91 \, \text{A} \).

Step 4: Calculate Total Resistance

The total resistance for the series circuit is the sum of all individual resistances:\[ R_{total} = 2.0 + 12.0 + 3.0 = 17.0 \, \Omega \].

Step 5: Calculate Maximum Voltage of Battery

Using Ohm's Law, \( V = I \times R \), where \( V \) is the voltage, \( I = 0.91 \, \text{A} \) is the maximum current determined earlier, and \( R_{total} = 17.0 \, \Omega \), calculate the maximum voltage:\[ V_{max} = 0.91 \times 17.0 = 15.47 \, \text{V} \].

Step 6: Calculate Power Delivered to Circuit

The power delivered by the battery can be found using the power formula, \( P = V \times I \):\[ P_{delivered} = 15.47 \times 0.91 = 14.08 \, \text{W} \].

Key Concepts

Ohm's Law

Ohm's Law is a fundamental principle in electrical circuits that relates voltage, current, and resistance. It is summarized by the formula:

• \( V = I \times R \)

Where:

• \( V \) is the voltage across the circuit (in volts).
• \( I \) is the current flowing through the circuit (in amperes).
• \( R \) is the resistance of the circuit (in ohms).

This simple formula helps us determine how much current flows through a resistor when the voltage is applied, or conversely, how much voltage is needed for a desired current in a circuit with a known resistance.
For example, in our exercise with a series circuit, we use Ohm's Law to compute the maximum voltage the battery can provide without causing any resistor to exceed its power rating. By finding the total resistance and applying the maximum allowable current, we determine the greatest voltage possible, which is crucial for safe circuit operation.

Power Rating

The power rating of a resistor is the maximum amount of energy it can safely dissipate as heat without being damaged. It is crucial to know the power rating when designing or analyzing circuits to prevent component failure.
Power is calculated using the formula:

• \( P = I^2 \times R \) or equivalently \( P = V^2 / R \)

Where:

• \( P \) is the power (in watts).
• \( I \) is the current (in amperes).
• \( V \) is the voltage (in volts).
• \( R \) is the resistance (in ohms).

In our exercise, we consider each resistor's power rating to ensure they do not exceed these limits. By calculating the maximum current each resistor can handle, we effectively protect them from burning or overloading. The lowest calculated current value dictates the maximum safe current for the entire series circuit.

Resistance

Resistance is the property of a material that opposes the flow of electric current. It is measured in ohms (\( \Omega \)).

Understanding resistance is fundamental in calculating how much current will pass through a circuit for a given voltage.

• In series circuits, like the one in our exercise, the total resistance is the sum of individual resistances:
• \( R_{total} = R_1 + R_2 + R_3 + \ldots \)

For our series circuit:

• \( R_{total} = 2.0 + 12.0 + 3.0 = 17.0 \Omega \)

This calculated resistance helps us in finding the maximum possible voltage supplied by a battery using Ohm's Law. The concept of resistance is also key in managing the power flow within a circuit, ensuring all components function within their prescribed limits.