Problem 71 - University Physics with Modern Physics

Question

A lightning bolt strikes one end of a steel lightning rod, producing a 15,000-A current burst that lasts for 65 \(\mu\)s. The rod is 2.0 m long and 1.8 cm in diameter, and its other end is connected to the ground by 35 m of 8.0-mm-diameter copper wire. (a) Find the potential difference between the top of the steel rod and the lower end of the copper wire during the current burst. (b) Find the total energy deposited in the rod and wire by the current burst.

Step-by-Step Solution

Answer

The potential difference is calculated using the combined resistance of the steel rod and copper wire. The energy is found using the formula for Joule heating with given current and total resistance.

Step 1: Understand the Problem and Given Data

We need to find two things:(a) The potential difference between the top of the steel rod and the lower end of the copper wire during the current burst.(b) The total energy deposited in the rod and wire by the current burst.Given data:- Current, \( I = 15000 \) A- Time duration, \( t = 65 \times 10^{-6} \) sSteel rod:- Length, \( L_1 = 2.0 \) m- Diameter, \( d_1 = 1.8 \times 10^{-2} \) m- Resistivity of steel (approx.), \( \rho_1 = 1.43 \times 10^{-7} \) \( \Omega \cdot \text{m} \)Copper wire:- Length, \( L_2 = 35 \) m- Diameter, \( d_2 = 8.0 \times 10^{-3} \) m- Resistivity of copper, \( \rho_2 = 1.68 \times 10^{-8} \) \( \Omega \cdot \text{m} \)

Step 2: Calculate Resistance of Steel Rod

First, calculate the cross-sectional area (A) of the steel rod:\[ A_1 = \frac{\pi d_1^2}{4} = \frac{\pi (1.8 \times 10^{-2})^2}{4} \]Using the formula for resistance:\[ R_1 = \frac{\rho_1 L_1}{A_1} \] Substitute the values to find \( R_1 \).

Step 3: Calculate Resistance of Copper Wire

Calculate the cross-sectional area (A) of the copper wire:\[ A_2 = \frac{\pi d_2^2}{4} = \frac{\pi (8.0 \times 10^{-3})^2}{4} \]Using the formula for resistance:\[ R_2 = \frac{\rho_2 L_2}{A_2} \] Substitute the values to find \( R_2 \).

Step 4: Calculate Total Resistance and Potential Difference

The total resistance across the rod and wire is:\[ R_{\text{total}} = R_1 + R_2 \]The potential difference, \( V \), is given by Ohm's Law:\[ V = I \times R_{\text{total}} \]Substitute the values of \( I \) and \( R_{\text{total}} \) to find \( V \).

Step 5: Calculate Total Energy Deposited

Energy deposited can be calculated using the formula:\[ E = I^2 \times R_{\text{total}} \times t \]Substitute the values of \( I, R_{\text{total}}, \) and \( t \) to find \( E \).

Step 6: Conclusion: Final Results

Based on the calculations, obtain the potential difference and total energy deposited for the given scenario.

Key Concepts

Resistivity

Resistivity is a fundamental property of materials that illustrates how much they resist the flow of electric current. It is a measure of how strongly a given material opposes the passage of electrons. The resistivity of a material is denoted by the symbol \( \rho \) and is measured in ohm-meters (\( \Omega \cdot \text{m} \)).
For instance, the resistivity of steel is approximately \( 1.43 \times 10^{-7} \Omega \cdot \text{m} \), while that of copper is \( 1.68 \times 10^{-8} \Omega \cdot \text{m} \). This means copper is a much better conductor than steel owing to its lower resistivity, allowing electric current to flow more easily through it.

• The formula to calculate resistivity is: \( \rho = R \times \frac{A}{L} \), where \( R \) is resistance, \( A \) is the cross-sectional area, and \( L \) is the length of the material.
• A higher resistivity means more energy is needed to push electrons through the material.

Understanding resistivity is crucial when determining the most efficient materials for electrical components, affecting their performance and energy efficiency.

Ohm's Law

Ohm's Law is one of the primary principles of electric circuits, fundamentally describing the relationship between voltage, current, and resistance. It is represented by the formula \( V = I \times R \). Here, \( V \) is the voltage (or potential difference), \( I \) is the current in amperes, and \( R \) is the resistance in ohms.
This law helps us understand how different components and conductors affect the flow of electricity. In the context of a lightning strike passing through a steel rod into a copper wire, Ohm's Law allows us to calculate the potential difference created along the rod and the wire.

• For a given resistance, increasing the current will proportionally increase the voltage.
• Understanding Ohm's Law is essential for safely designing and maintaining electrical circuits, as it predicts how much voltage is required to achieve a particular current flow.

By applying Ohm's Law, engineers can determine how much voltage a circuit component will require to function correctly or survive safely under different currents.

Potential Difference

Potential difference, often referred to as voltage, is the disparity in electric potential energy between two points in a circuit. It is the force that motivates electrons to travel from one point to another and is measured in volts (V).
In the scenario of a lightning bolt striking and current traveling down a rod and wire, potential difference plays a crucial role in determining the energy distribution across the circuit.

• Voltage can be thought of as electrical 'pressure' that pushes electrons through a conductor.
• Without a potential difference, electrons would not flow, rendering the circuit inactive.

This concept is essential in deciding how energy is allocated in circuits, and how effectively electric devices can operate under specific voltages.

Energy Deposition

Energy deposition in an electric circuit refers to the amount of electrical energy transformed into other forms of energy, such as heat, light, or motion, as current passes through a resistive element.
In a lightning rod and wire experiencing a current burst, significant energy is deposited due to the high current and inherent resistance of the materials.

• The energy deposited can be calculated using the formula: \( E = I^2 \times R \times t \), where \( E \) is energy in joules, \( I \) is current in amperes, \( R \) is resistance, and \( t \) is time in seconds.
• This energy conversion might result in heat generation, which could potentially damage the conductors or surrounding materials if not properly accounted for.

Calculating energy deposition is crucial in electrical system design to ensure the sustained efficiency and safety of the components under different operational conditions.