The girl in Figure 26.12 of the chapter has her hand on the...

The girl in Figure 26.12 of the chapter has her hand on the sphere of a Van de Graaff generator that is at a potential of $400,000 \mathrm{V}$. She is standing on an insulating platform, so no current flows through her. But what happens if she touches something that is grounded? Is she still safe? Figure $P 26.57$ shows the equivalent circuit. $C_{\mathrm{VDG}}=20 \mathrm{pF}$ represents the capacitance of the Van de Graaff sphere, $C_{\text {girl }}=100 \mathrm{pF}$ is the capacitance of the girl's body, and $R=5 \mathrm{k} \Omega$ is the resistance of her body, from one hand to the other. The switch is closed when she touches ground. What is the initial current at the instant the switch is closed? b. What is the time constant for the current to decay? c. Occupational safety experts have found that a shock is safe if the product of the voltage, the current, and the time the current is delivered is less than $13.5 \mathrm{V} \cdot \mathrm{A} \cdot \mathrm{s}=13.5 \mathrm{J}$ (see Problem 26.28). Estimate this product. Is this shock safe?

The girl in Figure 26.12 of the chapter has her hand on the sphere of a Van de Graaff generator that is at a potential of $400,000 \mathrm{V}$. She is standing on an insulating platform, so no current flows through her. But what happens if she touches something that is grounded? Is she still safe? Figure $P 26.57$ shows the equivalent circuit. $C_{\mathrm{VDG}}=20 \mathrm{pF}$ represents the capacitance of the Van de Graaff sphere, $C_{\text {girl }}=100 \mathrm{pF}$ is the capacitance of the girl's body, and $R=5 \mathrm{k} \Omega$ is the resistance of her body, from one hand to the other. The switch is closed when she touches ground.
What is the initial current at the instant the switch is closed?
b. What is the time constant for the current to decay?
c. Occupational safety experts have found that a shock is safe if the product of the voltage, the current, and the time the current is delivered is less than $13.5 \mathrm{V} \cdot \mathrm{A} \cdot \mathrm{s}=13.5 \mathrm{J}$ (see Problem 26.28). Estimate this product. Is this shock safe?

Key Concepts

Capacitance

Capacitance is a measure of a component’s ability to store an electric charge and energy in an electric field when a potential difference is applied. In circuits, capacitors are used to store energy and can affect transient responses when a circuit’s state changes, such as when a switch is closed.

Equivalent Circuit Modeling

An equivalent circuit is a simplified representation of a more complex system using idealized electrical components like resistors, capacitors, and voltage sources. This model helps in analyzing and understanding the behavior of the physical system, especially when it involves transient events such as sudden changes in connectivity.

RC Circuit Transients

RC circuits, which consist of resistors and capacitors, exhibit exponential charging and discharging behavior. When a sudden change, like closing a switch, occurs, the voltage across the capacitor and the current through the resistor change over time following an exponential law. This transient behavior is crucial for predicting the immediate response of the circuit.

Time Constant (? = RC)

The time constant in an RC circuit, defined as ? = RC, indicates the characteristic time over which the capacitor charges or discharges. It determines how quickly the current or voltage decays from its initial value. A short time constant implies a rapid decay, while a longer time constant means the transient persists for a longer period.

Initial Discharge Current in Capacitive Circuits

The initial current upon closing a switch in a capacitive circuit is determined by the voltage difference available and the resistance in the discharge path. At the moment of switching, the full voltage appears across the resistor, and Ohm’s law can be used directly to calculate the peak current, which then decays exponentially as the capacitor discharges.

Electrical Safety Assessment

Electrical safety assessments involve evaluating the product of voltage, current, and time to determine the energy delivered during an electrical shock. This energy, often expressed in joules, helps determine whether an incident poses a risk to human health. Understanding the transient discharge in circuits is key to assessing if the energy delivered is within safe limits.

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