Problem 79 - College Physics

Question

Struck by lightning. Lightning strikes can involve currents as high as \(25,000\) A that last for about 40\(\mu\) s. If a person is struck by a bolt of lightning with these properties, the current will pass through his body. We shall assume that his mass is 75 kg, that he is wet (after all, he is in a rainstorm) and therefore has a resistance of \(1.0 \mathrm{k} \Omega,\) and that his body is all water (which is reasonable for a rough, but plausible, approximation).(a) By how many degrees Celsius would this lightning bolt increase the temperature of 75 kg of water? (b) Given that the internal body temperature is about \(37^{\circ} \mathrm{C}\) , would the person's temperature actually increase that much? Why not? What would happen first?

Step-by-Step Solution

Answer

The temperature would increase by approximately 79.63°C, but severe damage would occur before this thermal increase.

Step 1: Calculate Energy Delivered

To find the energy delivered to the person's body, we use the formula for electrical energy: \[ E = I^2 \cdot R \cdot t \]where \( I = 25,000 \) A is the current, \( R = 1,000 \) \( \Omega \) (since 1 k\( \Omega = 1,000 \Omega \)) is the resistance, and \( t = 40 \mu s = 40 \times 10^{-6} \) s is the time duration. Let's calculate:\[ E = (25,000)^2 \cdot 1,000 \cdot 40 \times 10^{-6} \]\[ E = 25,000^2 = 625,000,000 \]\[ E = 625,000,000 \cdot 1,000 \cdot 40 \times 10^{-6} = 25,000,000 \text{ J} \]Thus, the energy delivered is 25,000,000 J.

Step 2: Calculate Temperature Increase

Now, we use the formula for heat transferred to a substance:\[ Q = mc\Delta T \]where \( Q = 25,000,000 \text{ J} \) from Step 1, \( m = 75 \text{ kg} \) is the mass of the water (person's body), and \( c = 4,186 \text{ J/kg°C} \) is the specific heat capacity of water. We solve for \( \Delta T \):\[ \Delta T = \frac{Q}{mc} = \frac{25,000,000}{75 \times 4,186} \]\[ \Delta T = \frac{25,000,000}{313950} \approx 79.63°C \]Thus, the temperature increase would be approximately 79.63°C.

Step 3: Analyze Physical Effects

Given the calculated increase of approximately 79.63°C, the internal body temperature would rise far beyond survivable levels (from 37°C to over 116°C). However, in reality, this increase would not occur because such a sudden and high magnitude current would cause immediate severe burns and other injuries, likely leading to immediate cardiac arrest or severe tissue damage before such a temperature rise could be achieved. Hence, the person would likely suffer fatal injuries from the electrical discharge rather than the thermal increase.

Key Concepts

Lightning Strike Analysis

When analyzing a lightning strike, we deal with colossal amounts of electrical energy being transferred in a very short timeframe. Lightning strikes can generate currents as high as 25,000 A, lasting merely for fractions of a second, around 40 microseconds ( ext{µs} ). To put this in perspective, household electrical currents are typically around 15 to 20 A. A lightning strike enters through one part of the body and exits through another, effectively using the human body as a conductor. The body's resistance, with an assumed wet external surface, is approximately 1 ext{k} Ω (1,000 Ω). Resistance is crucial because it determines how much of this mammoth current can flow through the body, impacting the energy that the body absorbs. Understanding these parameters helps us to calculate the potential energy absorbed by a person during a lightning strike. This involves breaking down the concept of electrical energy transfer into calculations that yield the magnitude of energy involved.

Specific Heat Capacity

Specific heat capacity is a fundamental concept in understanding how substances absorb and retain thermal energy. It represents the amount of energy required to raise the temperature of a unit mass of a substance by 1°C. For instance: - Water has a high specific heat capacity, about 4,186 ext{J/kg°C} , meaning it can absorb a lot of heat before its temperature increases significantly. When calculating the effects of a lightning strike on the body, we assume the body is essentially a mass of water. This simplification lets us use the specific heat capacity of water to determine how much the body’s temperature would rise given certain levels of absorbed energy. A higher specific heat capacity is protective in controlling drastic temperature changes, thereby delaying harmful effects due to thermal energy absorption. This property is crucial in the case of a lightning strike, as it helps predict potential internal temperature changes.

Thermal Energy Calculation

Thermal energy calculations provide insights into the temperature increase resulting from energy absorbed from a lightning strike. We utilize the thermal energy formula: \[ Q = mc\Delta T \]where \( Q \) is the thermal energy added, \( m \) is the mass (in this case, 75 kg for the human body assumed to be water), \( c \) is the specific heat capacity, and \( \Delta T \) is the temperature change.Plugging in the numbers obtained from the lightning calculation:- \( Q = 25,000,000 \) J (from the energy delivered in the lightning strike),- \( m = 75 \) kg,- \( c = 4,186 \text{ J/kg°C} \).The result is an estimated temperature rise of approximately 79.63°C. It is crucial to understand the real-world implications of this result. Despite the theoretical temperature rise, in practice, the body would experience immediate damage from the electrical current long before such a temperature shift occurred. This is due to the rapid and intense nature of the energy input which could cause burns and other severe injuries immediately upon exposure.