Question

A thundercloud similar to the one described in Example 24.3 produces a lightning bolt that strikes a radio tower. If the lightning bolt transfers \(5.00 \mathrm{C}\) of charge in about \(0.100 \mathrm{~ms}\) and the potential remains constant at \(70.0 \mathrm{MV}\), find (a) the average current, (b) the average power, (c) the total energy, and (d) the effective resistance of the air during the lightning strike.

Short answer

Answer: (a) The average current is 50,000 A, (b) the average power is 3.50x10^12 W, (c) the total energy is 3.50x10^8 J, and (d) the effective resistance of the air is 1.40x10^3 Ω.

Step-by-step solution

Calculate the average current

To calculate the average current, use the formula \(I = \frac{Q}{t}\). Given the transferred charge \(Q = 5.00 \mathrm{C}\) and time \(t = 0.100 \mathrm{ms} = 0.100\times10^{-3}\mathrm{s}\), we can calculate the average current as follows: \(I = \frac{5.00}{0.100\times10^{-3}} = 50,000 \mathrm{A}\) So, the average current during the lightning strike is \(50,000 \mathrm{A}\).

Calculate the average power

Now, we will calculate the average power using the formula \(P = VI\). We have the potential \(V = 70.0 \mathrm{MV} = 70.0\times10^{6}\mathrm{V}\) and average current \(I = 50,000 \mathrm{A}\). The average power is: \(P = (70.0\times10^{6})(50,000) = 3.50\times10^{12} \mathrm{W}\) The average power during the lightning strike is \(3.50\times10^{12} \mathrm{W}\).

Calculate the total energy

In order to calculate the total energy, we use the formula \(E = QV\). We have the transferred charge \(Q = 5.00 \mathrm{C}\) and potential \(V = 70.0 \mathrm{MV} = 70.0\times10^{6}\mathrm{V}\). Therefore, the total energy is: \(E = (5.00)(70.0\times10^{6}) = 3.50\times10^{8} \mathrm{J}\) The total energy during the lightning strike is \(3.50\times10^{8} \mathrm{J}\).

Calculate the effective resistance

Finally, we will calculate the effective resistance using the formula \(R = \frac{V}{I}\). We have the potential \(V = 70.0 \mathrm{MV} = 70.0\times10^{6}\mathrm{V}\) and the average current \(I = 50,000 \mathrm{A}\). The effective resistance of the air during the lightning strike is: \(R = \frac{70.0\times10^{6}}{50,000} = 1.40\times10^{3} \mathrm{\Omega}\) The effective resistance of the air during the lightning strike is \(1.40\times10^{3} \mathrm{\Omega}\). In conclusion, during the lightning strike, (a) the average current is \(50,000 \mathrm{A}\), (b) the average power is \(3.50\times10^{12} \mathrm{W}\), (c) the total energy is \(3.50\times10^{8} \mathrm{J}\), and (d) the effective resistance of the air is \(1.40\times10^{3} \mathrm{\Omega}\).

Key concepts

Average Current

The concept of average current represents the steady flow of electric charge over a specific period of time. It tells us how much charge is moving through a circuit on average in a given timeframe. To calculate the average current, we use the formula: \[ I = \frac{Q}{t} \] where:

• \( I \) is the average current in amperes (A)
• \( Q \) is the total charge transferred in coulombs (C)
• \( t \) is the time period in seconds (s)

In the context of a lightning strike, knowing the average current is crucial as it indicates the magnitude of charge flow between clouds and the ground. High current can lead to significant damage.
For the given problem, with \( Q = 5.00 \) C of charge transferred in \( 0.100 \times 10^{-3} \) s, the average current comes out to \( 50,000 \) A.

Average Power

Average power is a measure of how much energy is used or transferred per unit of time. In the field of electricity and magnetism, it's commonly described using the formula:\[ P = VI \]where:

• \( P \) is the power in watts (W)
• \( V \) is the potential difference in volts (V)
• \( I \) is the current in amperes (A)

Understanding average power is important because it shows us how fast electrical energy is being converted. In lightning, it gives an idea of the tremendous energy involved. For the specific situation with \( V = 70.0 \times 10^{6} \) V and \( I = 50,000 \) A, the average power is calculated as \( 3.50 \times 10^{12} \) W.
This immense power level underscores the raw energy of a lightning strike.

Energy Calculation

Calculating the energy in an electrical system involves figuring out how much work is done or energy is transferred by the system. The formula used is:\[ E = QV \]where:

• \( E \) is the energy in joules (J)
• \( Q \) is the charge in coulombs (C)
• \( V \) is the potential difference in volts (V)

In the case of a lightning strike, the energy calculation gives insight into how much energy is being released instantaneously.
With values \( Q = 5.00 \) C and \( V = 70.0 \times 10^{6} \) V, the energy discharged is \( 3.50 \times 10^{8} \) J. This showcases the power of natural electricity events, given the vast amount of energy within a fraction of a second.

Resistance Calculation

Resistance is a measure of how much a material opposes the flow of electric current. It indicates how difficult it is for electricity to pass through a conductor. The formula used to determine resistance is:\[ R = \frac{V}{I} \]where:

• \( R \) is the resistance in ohms (\( \Omega \))
• \( V \) is the voltage in volts (V)
• \( I \) is the current in amperes (A)

Understanding resistance is important as it determines how much current will flow under a given voltage. For the thundercloud example, resistance explains how the air acts as a barrier to electrical flow.
With \( V = 70.0 \times 10^{6} \) V and \( I = 50,000 \) A, the effective resistance is calculated to be \( 1.40 \times 10^{3} \Omega \). This resistance measure helps clarify the conditions under which the lightning occurs.